Rational Functions and Equations

Transcription

1 Rational Functions and Equations 12A Rational Functions and Epressions Lab Model Inverse Variation 12-1 Inverse Variation 12-2 Rational Functions 12-3 Simplifying Rational Epressions Lab Graph Rational Functions 12B Operations with Rational Epressions and Equations 12-4 Multiplying and Dividing Rational Epressions 12-5 Adding and Subtracting Rational Epressions Lab Model Polynomial Division 12-6 Dividing Polynomials 12-7 Solving Rational Equations Et Trigonometric Ratios By DES IGN Ratios and rational epressions can be used to eplore perspective in art and dimensions in package design. Try your hand at both. KEYWORD: MA7 ChProj 846 Chapter 12

2 Vocabulary Match each term on the left with a definition on the right. 1. perfect-square A. the greatest factor that is shared by two or more terms trinomial 2. greatest common factor 3. monomial 4. polynomial 5. reciprocals B. a number, a variable, or a product of numbers and variables with whole-number eponents C. two numbers whose product is 1 D. a polynomial with three terms E. the sum or difference of monomials F. a trinomial that is the result of squaring a binomial Simplify Fractions Simplify _ _ _ _ 66 Add and Subtract Fractions Add or subtract _ _ 8 - _ _ 4 + _ _ _ 9 + _ _ 1 3 Factor GCF from Polynomials Factor each polynomial Properties of Eponents Simplify each epression jk a 3 3 a ab 4 a 2 b y y 27. a 2 b 3ab rs 3rs m 2 n 2 4mn 2 Simplify Polynomial Epressions Simplify each epression y - 8y 31. 2r - 4s + 3s - 8r 32. ab 2 - ab + 4ab a 2 b + a 2 b g (g - 4) + g 2 + g Rational Functions and Equations 847

3 Previously you identified, wrote, and graphed equations of direct variation. identified and graphed quadratic, eponential, and square-root functions. used factoring to solve quadratic equations. simplified radical epressions and solved radical equations. Key Vocabulary/Vocabulario asymptote discontinous function ecluded values inverse variation rational equation rational epression rational function asíntota función discontinua valores ecluidos variación inversa ecuación racional epresión racional función racional Vocabulary Connections You will study how to identify, write, and graph equations of inverse variation. how to graph rational functions and simplify rational epressions. how to solve rational equations. You can use the skills in this chapter to build upon your knowledge of graphing and transforming various types of functions. to solve problems involving inverse variation in classes such as Physics and Chemistry. to calculate costs when working with a fied budget. To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. What are some other words that mean the same as continuous? The prefi dis- generally means not. Describe what the graph of a discontinuous function might look like. 2. What does it mean for someone or something to be included in a group? What about ecluded? What might it mean for some values to be ecluded values for a particular function? 3. A direct variation is a relationship between two variables, and y, that can be written in the form y = k where k is a nonzero constant. The inverse of a number is 1. Use this information to write the form of an inverse variation. 4. You learned in Chapter 1 that an algebraic epression is an epression that contains one or more variables, numbers, or operations. You also learned that a rational number is a number that can be written in the form of a fraction. Combine these terms to define rational epression. Give an eample. 848 Chapter 12

4 Study Strategy: Prepare for Your Final Eam Math is a cumulative subject, so your final eam will probably cover all of the material you have learned since the beginning of the course. Preparation is essential for you to be successful on your final eam. It may help you to make a study timeline like the one below. 2 weeks before the final: Look at previous eams and homework to determineareasineedtofocuson;rework need to on; rework problems that were incorrect or incomplete. Make a list of all formulas, postulates, and theoremsineedtoknowforthefinal. to final. Create a practice eam using problems from the book that are similar to problems from each eam. 1 week before the final: Take the practice eam and check it. For each problem I miss, find 2 or 3 similar ones and work those. Work with a friend in the class to quiz each other on formulas, postulates, and theorems from my list. 1 day before the final: Make sure I have pencils, calculator (check batteries!), ruler, compass, and protractor. Try This 1. Create a timeline that you will use to study for your final eam. Rational Functions and Equations 849

5 12-1 Model Inverse Variation The relationship between the width and the length of a rectangle with a constant area is an inverse variation. In this activity, you will study this relationship by modeling rectangles with square tiles or grid paper. Use with Lesson 12-1 Activity Use 12 square tiles to form a rectangle with an area of 12 square units, or draw the rectangle on grid paper. Use a width of 1 unit and a length of 12 units. Your rectangle should look like the one shown. Using the same 12 square tiles, continue forming rectangles by changing the width and length until you have formed all the different rectangles you can that have an area of 12 square units. Copy and complete the table as you form each rectangle. Width () Length (y) Area ( y) Plot the ordered pairs from the table on a graph. Draw a smooth curve through the points. Try This 1. Look at the table and graph above. What happens to the length as the width increases? Why? 2. This type of relationship is called an inverse variation. Why do you think it is called that? 3. For each point, what does y equal? Complete the equation y =. Solve this equation for y. 4. Form all the different rectangles that have an area of 24 square units. Record their widths and lengths in a table. Graph your results. Write an equation relating the width and length y. 5. Make a Conjecture Using the equations you wrote in 3 and 4, what do you think the equation of any inverse variation might look like when solved for y? 850 Chapter 12 Rational Functions and Equations

6 12-1 Inverse Variation Objective Identify, write, and graph inverse variations. Vocabulary inverse variation A direct variation is an equation that can be written in the form y = k, where k is a nonzero constant. Why learn this? Inverse variation can be used to find the frequency at which a guitar string vibrates. (See Eample 3.) A relationship that can be written in the form y = k, where k is a nonzero constant and 0, is an inverse variation. The constant k is the constant of variation. Inverse variation implies that one quantity will increase while the other quantity will decrease (the inverse, or opposite, of increase). Multiplying both sides of y = k by gives y = k. So, for any inverse variation, the product of and y is a nonzero constant. Inverse Variations WORDS NUMBERS ALGEBRA y varies inversely as. y is inversely proportional to. y = 3 _ y = 3 y = k _ y = k(k 0) There are two methods to determine whether a relationship between data is an inverse variation. You can write a function rule in y = k form, or you can check whether y is constant for each ordered pair. EXAMPLE 1 Identifying an Inverse Variation Tell whether each relationship is an inverse variation. Eplain. A y Method 1 Write a function rule. y = _ 20 Can write in y = k form. The relationship is an inverse variation. Method 2 Find y for each ordered pair. 1(20) = 20, 2 (10) = 20, 4 (5) = 20 The product y is constant, so the relationship is an inverse variation. B y Method 1 Write a function rule. y = 3 Cannot write in y = k form. The relationship is not an inverse variation. Method 2 Find y for each ordered pair. 2(6) = 12, 3 (9) = 27, 6 (18) = 108 The product y is not constant, so the relationship is not an inverse variation Inverse Variation 851

7 Tell whether each relationship is an inverse variation. Eplain. C 5y =-21 _ 5y _ 5 = -21 Find y. Since y is multiplied by 5, divide both sides by 5 5 to undo the multiplication. y = _ -21 Simplify. 5 y equals the constant -21, so the relationship is an inverse variation. 5 Tell whether each relationship is an inverse variation. Eplain. 1a. y b. y c. 2 + y = 10 Since k is a nonzero constant, y 0. Therefore, neither nor y can equal 0, and no solution points will be on the - or y-aes. An inverse variation can also be identified by its graph. Some inverse variation graphs are shown. Notice that each graph has two parts that are not connected. Also notice that none of the graphs contain (0, 0). This is because (0, 0) can never be a solution of an inverse variation equation. EXAMPLE 2 Graphing an Inverse Variation Write and graph the inverse variation in which y = 2 when = 4. Step 1 Find k. k = y Write the rule for constant of variation. = 4(2) Substitute 4 for and 2 for y. = 8 Step 2 Use the value of k to write an inverse variation equation. y = k _ y = 8 _ Write the rule for inverse variation. Substitute 8 for k. Step 3 Use the equation to make a table of values y undef Step 4 Plot the points and connect them with smooth curves. 2. Write and graph the inverse variation in which y = 1 2 when = Chapter 12 Rational Functions and Equations

8 EXAMPLE 3 Music Application The inverse variation y = 2400 relates the vibration frequency y in hertz (Hz) to the length in centimeters of a guitar string. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate the frequency of vibration when the string length is 100 centimeters. Step 1 Solve the function for y so you can graph it. y = 2400 y = 2400 _ Divide both sides by. Step 2 Decide on a reasonable domain and range. > 0 y > 0 Length is never negative and 0. Because and y are both positive, y is also positive. Recall that sometimes domain and range are restricted in realworld situations. Step 3 Use values of the domain to generate reasonable ordered pairs y Step 4 Plot the points. Connect them with a smooth curve. Step 5 Find the y-value where = 100. When the string length is 100 cm, the frequency of vibration is about 24 Hz. 3. The inverse variation y = 100 represents the relationship between the pressure in atmospheres (atm) and the volume y in mm 3 of a certain gas. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate the volume of the gas when the pressure is 40 atmospheric units. The fact that y = k is the same for every ordered pair in any inverse variation can help you find missing values in the relationship. Product Rule for Inverse Variation If ( 1, y 1) and ( 2, y 2) are solutions of an inverse variation, then 1 y 1 = 2 y 2. EXAMPLE 4 Using the Product Rule Let 1 = 3, y 1 = 2, and y 2 = 6. Let y vary inversely as. Find 2. 1 y 1 = 2 y 2 (3)(2) = 2 (6) Write the Product Rule for Inverse Variation. Substitute 3 for 1, 2 for y 1, and 6 for y 2. 6 = = = 2 Simplify. Solve for 2 by dividing both sides by 6. Simplify Inverse Variation 853

9 4. Let 1 = 2, y 1 =-6, and 2 =-4. Let y vary inversely as. Find y 2. EXAMPLE 5 Physics Application In Eample 5, 1 and y 1 represent volume and pressure before the handle is pushed in, and 2 and y 2 represent volume and pressure after the handle is pushed in. Boyle s law states that the pressure of a quantity of gas varies inversely as the volume of the gas y. The volume of air inside a bicycle pump is 5.2 in 3, and the pressure is 15.5 psi. Assuming no air escapes, what is the pressure of the air inside the pump after the handle is pushed in, and the air is compressed to a volume of 2.6 in 3? 1 y 1 = 2 y 2 Use the Product Rule for Inverse Variation. (5.2)(15.5) = (2.6)y 2 Substitute 5.2 for 1, 15.5 for y 1, and 2.6 for = 2.6 y 2 Simplify. _ 80.6 _ Solve for y 2 by dividing both sides by = 2.6y = y 2 Simplify. The pressure after the handle is pushed in is 31 psi. 5. On a balanced lever, weight varies inversely as the distance from the fulcrum to the weight. The diagram shows a balanced lever. How much does the child weigh? THINK AND DISCUSS 1. Name two ways you can identify an inverse variation. 2. GET ORGANIZED Copy and complete the graphic organizer. In each bo, write an eample of the parts of the given inverse variation. 854 Chapter 12 Rational Functions and Equations

10 12-1 Eercises GUIDED PRACTICE 1. Vocabulary Describe the graph of an inverse variation. KEYWORD: MA KEYWORD: MA7 Parent SEE EXAMPLE 1 p. 851 Tell whether each relationship is an inverse variation. Eplain y y y = y = 3 SEE EXAMPLE 2 p. 852 SEE EXAMPLE 3 p. 853 SEE EXAMPLE 4 p. 853 SEE EXAMPLE 5 p Write and graph the inverse variation in which y = 2 when = Write and graph the inverse variation in which y = 6 when = Travel The inverse variation y = 30 relates the constant speed in mi/h to the time y in hours that it takes to travel 30 miles. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate how many hours it would take traveling 4 mi/h. 9. Let 1 = 3, y 1 = 12, and 2 = 9. Let y vary inversely as. Find y Let 1 = 1, y 1 = 4, and y 2 = 16. Let y vary inversely as. Find Mechanics The rotational speed of a gear varies inversely as the number of teeth on the gear. A gear with 12 teeth has a rotational speed of 60 rpm. How many teeth are on a gear that has a rotational speed of 45 rpm? Independent Practice For See Eercises Eample Etra Practice Skills Practice p. S26 Application Practice p. S39 PRACTICE AND PROBLEM SOLVING Tell whether each relationship is an inverse variation. Eplain y = 13 _ y 15. y = y Write and graph the inverse variation in which y =-2 when = Write and graph the inverse variation in which y =-6 when = Engineering The inverse variation y = 12 relates the current in amps to the resistance y in ohms of a circuit attached to a 12-volt battery. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate the resistance of a circuit when the current is 5 amps. 19. Let 1 =-3, y 1 =-4, and y 2 = 6. Let y vary inversely as. Find Let 1 = 7, y 1 = 9, and 2 = 6. Let y vary inversely as. Find y Inverse Variation 855

11 Winter Sports Snowshoes were originally made of wooden frames strung with animal intestines. Modern snowshoes are made with steel frames and may have cleats for gripping the snow. 21. Home Economics The length of fabric that June can afford varies inversely as the price per yard of the fabric. June can afford eactly 5 yards of fabric that costs $10.50 per yard. How many yards of fabric that costs $4.25 per yard can June buy? (Assume that she can only buy whole yards.) 22. Winter Sports When a person is snowshoeing, the pressure on the top of the snow in psi varies inversely as the area of the bottom of the snowshoe in square inches. The constant of variation is the weight of the person wearing the snowshoes in pounds. a. Helen weighs 120 pounds. About how much pressure does she put on top of the snow if she wears snowshoes that cover 360 in 2? b. Ma weighs 207 pounds. If he eerts 0.4 psi of pressure on top of the snow, what is the area of the bottom of his snowshoes in square inches? Determine if each equation represents a direct variation, an inverse variation, or neither. Find the constant of variation when one eists. 23. y = y = _ y = _ y = _ y = 4 _ y = _ y = _ y = Multi-Step A track team is competing in a 10 km race. The distance will be evenly divided among the team members. Write an equation that represents the distance d each runner will run if there are n runners. Does this represent a direct variation, inverse variation, or neither? Determine whether each data set represents a direct variation, an inverse variation, or neither y y y Multi-Step Your club awards one student a $2000 scholarship each year, and each member contributes an equal amount. Your contribution y depends on the number of members. Write and graph an inverse variation equation that represents this situation. What are a reasonable domain and range? 36. Estimation Estimate the value of y if y is inversely proportional to, = 4, and the constant of variation is 6π. 37. Critical Thinking Why will the point (0, 0) never be a solution to an inverse variation? 38. Write About It Eplain how to write an inverse variation equation of the form y = k when values of and y are known. 39. Write About It List all the mathematical terms you know that contain the word inverse. How are these terms all similar? How is inverse variation similar to these terms? 40. This problem will prepare you for the Multi-Step Test Prep on page 876. The total number of workdays it takes to build the frame of a house varies inversely as the number of people working in a crew. Let be the number of people in the crew and let y be the number of workdays. a. Find the constant of variation when y = 75 and = 2. b. Write the rule for the inverse of variation equation. c. Graph the equation of this inverse variation. 856 Chapter 12 Rational Functions and Equations

12 41. Which equation best represents the graph? y =-_ 1 4 y =-4 _ y = _ 1 4 y = _ Determine the constant of variation if y varies inversely as and y = 2 when = 7. 2_ 7 7_ Which of the following relationships does NOT represent an inverse variation? y y y = _ = y 44. Gridded Response At a carnival, the number of tickets Brad can buy is inversely proportional to the price of the tickets. He can afford 12 tickets that cost $2.50 each. How many tickets can Brad buy if each costs $3.00? CHALLENGE AND EXTEND 45. The definition of inverse variation says that k is a nonzero constant. What function would y = k represent if k = 0? 46. Mechanics A part of a car s braking system uses a lever to multiply the force applied to the P brake pedal. The force at the end of a lever varies inversely with the Fulcrum distance from the fulcrum. Point P is the end of the lever. A force of Brake pedal 2 lb is applied to the brake pedal. What is the force created at the point P? 47. Communication The strength of a radio signal varies inversely with the square of the distance from the transmitter. A signal has a strength of 2000 watts when it is 4 kilometers from the transmitter. What is the strength of the signal 6 kilometers from the transmitter? SPIRAL REVIEW Find the domain and range for each relation. Tell whether the relation is a function. (Lesson 4-2) 48. (-2, -4), (-2, -2), (-2, 0), (-2, 2) 49. (-4, 5), (-2, 3), (0, 1), (2, 3), (4, 5) Solve by completing the square. (Lesson 9-8) = d 2-6d - 7 = y 2 + 6y =- 5 _ A rectangle has a length of 6 cm and a width of 2 cm. Find the length of the diagonal and write it as a simplified radical epression. (Lesson 11-6) 6 in. 3 ft 12-1 Inverse Variation 857

13 12-2 Rational Functions Objectives Identify ecluded values of rational functions. Graph rational functions. Vocabulary rational function ecluded value discontinuous function asymptote Who uses this? Gemologists can use rational functions to maimize reflected light. (See Eample 4.) A rational function is a function whose rule is a quotient of polynomials in which the denominator has a degree of at least 1. In other words, there must be a variable in the denominator. The inverse variations you studied in the previous lesson are a special type of rational function. Rational functions: y = 2_, y = 3_ 4-2, y = 2 Not rational functions: y = _ 4, y = 3 For any function involving and y, an ecluded value is any -value that makes the function value y undefined. For a rational function, an ecluded value is any value that makes the denominator equal 0. EXAMPLE 1 Identifying Ecluded Values Identify the ecluded value for each rational function. A y = 8_ = 0 Set the denominator equal to 0. The ecluded value is 0. B y = 3_ = 0 Set the denominator equal to 0. =-3 Solve for. The ecluded value is -3. Identify the ecluded value for each rational function. 1a. y = _ 10 1b. y = _ 4 1c. y =-_ Most rational functions are discontinuous functions, meaning their graphs contain one or more jumps, breaks, or holes. This occurs at an ecluded value. One place that a graph of a rational function is discontinuous is at an asymptote. An asymptote is a line that a graph gets closer to as the absolute value of a variable increases. In the graph shown, both the - and y-aes are asymptotes. The graphs of rational functions will get closer and closer to but never touch the asymptotes. 858 Chapter 12 Rational Functions and Equations

14 Vertical lines are written in the form = b, and horizontal lines are written in the form y = c. For rational functions, vertical asymptotes will occur at ecluded values. Look at the graph of y = 1. The denominator is 0 when = 0 so 0 is an ecluded value. This means there is a vertical asymptote at = 0. Notice the horizontal asymptote at y = 0. 1 Look at the graph of y = + 2. Notice that - 3 the graph of the parent function y = 1 has been translated 3 units right and there is a vertical asymptote at = 3. The graph has also been translated 2 units up and there is a horizontal asymptote at y = 2. These translations lead to the following formulas for identifying asymptotes in rational functions. Identifying Asymptotes WORDS EXAMPLES A rational function in the form y = _ a - b + c has a vertical asymptote at the ecluded value, or = b, and a horizontal asymptote at y = c. y = 2 _ = 2 _ Vertical asymptote: = 0 Horizontal asymptote: y = 0 y = _ = - (-2) + 4 Vertical asymptote: = -2 Horizontal asymptote: y = 4 EXAMPLE 2 Identifying Asymptotes Identify the asymptotes. A y = - 6 Step 1 Write in y = - b + c form. y = Step 2 Identify the asymptotes. vertical: = 6 horizontal: y = 0 B y = 2_ Step 1 Identify the vertical asymptote = 0 Find the ecluded value. Set the denominator equal to Add 10 to both sides. 3 = 10 = 10_ 3 Solve for is an ecluded value Rational Functions 859

15 Step 2 Identify the horizontal asymptote. c = -7-7 can be written as + (-7) y = -7 y = c vertical asymptote: = 10 ; horizontal asymptote: y =-7 3 Identify the asymptotes. 2a. y = 2_ 2b. y = c. y = 3_ a To graph a rational function in the form y = + c when a = 1, you can graph - b the asymptotes and then translate the parent function y = 1. However, if a 1, the graph is not a translation of the parent function. In this case, you can use the asymptotes and a table of values. EXAMPLE 3 Graphing Rational Functions Using Asymptotes Graph each function. A y = 2_ + 1 Since the numerator is not 1, use the asymptotes and a table of values. Step 1 Identify the vertical and horizontal asymptotes. vertical: =-1 Use = b. + 1 = - (-1), so b = -1. horizontal : y = 0 Use y = c. c = 0 Step 2 Graph the asymptotes using dashed lines. Step 3 Make a table of values. Choose -values on both sides of the vertical asymptote y Step 4 Plot the points and connect them with smooth curves. The curves will get very close to the asymptotes, but will not touch them Since the numerator is 1, use the asymptotes and translate y = 1. B y = Step 1 Identify the vertical and horizontal asymptotes. vertical: = 2 Use = b. b = 2 horizontal: y =-4 Use y = c. c = -4 Step 2 Graph the asymptotes using dashed lines. Step 3 Draw smooth curves to show the translation. Graph each function. 3a. y = b. y = 2_ Chapter 12 Rational Functions and Equations

16 4 Gemology Application Some diamonds are cut using ratios calculated by a mathematician, Marcel Tolkowsky, in The amount of light reflected up through the top of a diamond (brilliancy) can be maimized using the ratio between the width of the diamond and the depth of the diamond. A gemologist has a diamond with a width of 90 millimeters. If represents 90 the depth of the diamond, then y = represents the brilliancy ratio y. Too deep Too shallow Ideal a. Describe the reasonable domain and range values. Both the depth of the diamond and the brilliancy ratio will be nonnegative, so nonnegative values are reasonable for the domain and range. b. Graph the function. Step 1 Identify the vertical and horizontal asymptotes. vertical: =0 horizontal: y = 0 Use = b. b = 0 Use y = c. c = 0 Step 2 Graph the asymptotes using dashed lines. The asymptotes will be the - and y-aes. Step 3 Since the domain is restricted to nonnegative values, only choose -values on the right side of the vertical asymptote. Depth of Diamond (mm) Brilliancy Ratio À > VÞÊ vê>ê > `Ê ÕÌ Step 4 Plot the points and connect them with smooth curves. À > VÞÊÀ>Ì EXAMPLE {ä Îä Óä ä ä ä Óä Îä {ä i«ì Ê 4. A librarian has a budget of $500 to buy copies of a software program. She will receive 10 free copies when she sets up an account with the supplier. The number of copies y of the 500 program that she can buy is given by y = _ + 10, where is the price per copy. a. Describe the reasonable domain and range values. b. Graph the function Rational Functions 861

17 The table shows some of the properties of the four families of functions you have studied and their graphs. Families of Functions LINEAR FUNCTION y = m + b Parent function: y = m is the slope. It rotates the graph about (0, b). b is the y-intercept. It translates the graph of y = vertically. QUADRATIC FUNCTION y = a 2 + b + c Parent function: y = 2 a determines the width of the parabola and the direction it opens. c translates the graph of y = a 2 vertically. The ais of symmetry is the vertical line =- b 2a. SQUARE-ROOT FUNCTION y = - a + b Parent function: y = a translates the graph of y = horizontally. b translates the graph of y = vertically. RATIONAL FUNCTION y = 1 _ - b + c Parent function: y = 1 b translates the graph y = 1 horizontally. c translates the graph of y = 1 vertically. THINK AND DISCUSS 1 1. Does y = have any ecluded values? Eplain Tell how to find the vertical and horizontal asymptotes of y = GET ORGANIZED Copy and complete the graphic organizer. In each bo, find the asymptotes for the given rational function. 862 Chapter 12 Rational Functions and Equations

18 12-2 Eercises KEYWORD: MA GUIDED PRACTICE KEYWORD: MA7 Parent 1. Vocabulary An -value that makes a function undefined is a(n)?. (asymptote or ecluded value) SEE EXAMPLE 1 p. 858 Identify the ecluded value for each rational function. 2. y = 4 _ 3. y = 2_ y =- 2 _ 5. y = 16_ - 4 SEE EXAMPLE 2 p. 859 Identify the asymptotes. 6. y = y = 4_ y = 2_ y = SEE EXAMPLE 3 p. 860 Graph each function. 10. y = 2_ y = y = _ y = SEE EXAMPLE 4 p. 861 Independent Practice For See Eercises Eample Etra Practice Skills Practice p. S26 Application Practice p. S Catering A caterer has $100 in her budget for fruit. Slicing and delivery of each pound of fruit costs $5. If represents the cost per pound of the fruit itself, then y = _ 100 represents the number of pounds y she can buy. + 5 a. Describe the reasonable domain and range values. b. Graph the function. PRACTICE AND PROBLEM SOLVING Identify the ecluded value for each rational function. 15. y = _ y = 17. y =- _ y = 12_ - 5 Identify the asymptotes. 19. y = 9_ 20. y = 2_ y = 7_ y = 7_ Graph each function. 23. y = 5_ 24. y = y = 26. y = Business A wholesaler is buying auto parts. He has $200 to spend. He receives 5 parts free with the order. The number of parts y he can buy, if the average price of the parts is dollars, is y = _ a. Describe the reasonable domain and range values. b. Graph the function. Find the ecluded value for each rational function. 28. y = _ y = 30. y = 2_ y = 3_ Graph each rational function. Show the asymptotes. 32. y = Multi-Step The function y = y = 2 _ y = 3_ y = relates the luminescence in lumens y of a 60-watt 2 lightbulb viewed from a distance of ft. Graph the function. Use the graph to find the luminescence of a 60-watt lightbulb viewed from a distance of 6 ft Rational Functions 863

19 Identify the asymptotes of each rational function. 37. y = _ y = _ y = _ y = _ Match each graph with one of the following functions. A. y = B. y = C. y = /ERROR ANALYSIS / In finding the horizontal asymptote of y = + 2-3, student A said the asymptote is at y =-3, and student B said it is at y =-2. Who is incorrect? Eplain the error. 45. Finance The time in months y that it will take to pay off a bill of $1200, when dollars are paid each month and the finance charge is $15 per month, is y = _ Describe the reasonable - 15 domain and range values and graph the function. 46. The table shows how long it takes different size landscaping teams to complete a project. a. Graph the data. b. Write a rational function to represent the data. c. How many hours would it take 12 landscapers to complete the project? Graph each function. Compare its graph to the graph of y = y = 1 _ y = 1 _ y = 1 _ y = 1 _ Find the domain that makes the range positive. 51. y = 10_ y = 10_ y = 5_ y = 55. Critical Thinking In which quadrants would you find the graph of y = a positive? when a is negative? 4_ 3-7 when a is 56. This problem will prepare you for the Multi-Step Test Prep on page 876. It takes a total of 250 workdays to build a house for charity. For eample, if 2 workers build the house, it takes them 125 actual construction days. If 10 workers are present, it takes 25 construction days to build the house. a. Write a function that represents the number of construction days to build as a function of the number of workers. b. What is the domain of this function? c. Sketch a graph of the function. 864 Chapter 12 Rational Functions and Equations

20 57. Write About It Graph each pair of functions on a graphing calculator. Then make a conjecture about the relationship between the graphs of the rational functions y = k -k and y = _. a. y = _ 1 ; y = _ -1 b. y = _ 3 ; y = _ -3 c. y = _ 5 ; y = _ Which function is graphed? y = _ y = _ y = _ y = _ Which rational function has a graph with the horizontal asymptote y =-1? y = _ -1 y = _ y = _ 1 y = _ Short Response Write a rational function whose graph is the same shape as the graph of f() = 1, but is translated 2 units left and 3 units down. Graph the function. CHALLENGE AND EXTEND Graph the equation y = _ a. Does this equation represent a rational function? Eplain. b. What is the domain of the function? c. What is the range of the function? d. Is the graph discontinuous? ( - 3)( - 1) 62. Graphing Calculator Are the graphs of f () = and g() = - 1 ( - 3) identical? Eplain. (Hint: Are there any ecluded values?) 63. Critical Thinking Write the equation of the rational function that has a horizontal asymptote at y = 3 and a vertical asymptote at =-2 and contains the point (1, 4). SPIRAL REVIEW Solve each equation by graphing the related function. (Lesson 9-5) = = = In the first five stages of a fractal design, a line segment has the following lengths, in centimeters: 240, 120, 60, 30, 15. Use this pattern and your knowledge of geometric sequences to determine the length of the segment in the tenth stage. (Lesson 11-1) Determine whether each function represents an inverse variation. Eplain. (Lesson 12-1) y = y =-4 y Rational Functions 865

21 12-3 Simplifying Rational Epressions Objectives Simplify rational epressions. Identify ecluded values of rational epressions. Vocabulary rational epression Why learn this? The shapes and sizes of plants and animals are partly determined by the ratio of surface area to volume. If an animal s body is small and its surface area is large, the rate of heat loss will be high. Hummingbirds must maintain a high metabolism to compensate for the loss of body heat due to having a high surfacearea-to-volume ratio. Formulas for surface-area-tovolume ratios are rational epressions. A rational epression is an algebraic epression whose numerator and denominator are polynomials. The value of the polynomial epression in the denominator cannot be zero since division by zero is undefined. This means that rational epressions may have ecluded values. EXAMPLE 1 Identifying Ecluded Values To review the Zero Product Property, see Lesson 9-6. To review factoring trinomials, see Chapter 8. Find any ecluded values of each rational epression. A B C 5_ 8r 8r = 0 Set the denominator equal to 0. r= _ 0 8 = 0 The ecluded value is 0. _ 9d + 1 d 2-2d Solve for r by dividing both sides by 8. d 2-2d = 0 d(d - 2) = 0 Set the denominator equal to 0. Factor. d = 0 or d - 2 = 0 d = 2 Use the Zero Product Property. Solve for d. The ecluded values are 0 and = 0 ( + 3)( + 2) = 0 Set the denominator equal to 0. Factor. + 3 = 0 or + 2 = 0 Use the Zero Product Property. = -3 or = -2 Solve each equation for. The ecluded values are -3 and -2. Find any ecluded values of each rational epression. 1a. 12_ 1b. 3b_ 1c. 3k 2 t + 5 b 2 + 5b k 2 + 7k Chapter 12 Rational Functions and Equations

22 A rational epression is in its simplest form when the numerator and denominator have no common factors ecept 1. Remember that to simplify fractions you can divide out common factors that appear in both the numerator and the denominator. You can do the same to simplify rational epressions. EXAMPLE 2 Simplifying Rational Epressions Be sure to use the original denominator when finding ecluded values. The ecluded values may not be seen in the simplified denominator. Simplify each rational epression, if possible. Identify any ecluded values. 3t A _ 3 12t _ 3t 2 Factor t Divide out common factors. Note that if t = 0, the _ 13 t 32 epression is undefined t 1 B C t 2 _ 4 ; t 0 _ _ 3( - 3) ( - 3) 1 _ ; 3 c_ c + 5 c_ c + 5 ; c -5 Simplify. The ecluded value is 0. Factor the numerator. Divide out common factors. Note that if = 3, the epression is undefined. Simplify. The ecluded value is 3. The numerator and denominator have no common factors. The ecluded value is -5. Simplify each rational epression, if possible. Identify any ecluded values. 2a. _ 5m 2 15m 2b. _ 6p p p c. 3n_ n - 2 From now on in this chapter, you may assume that the values of the variables that make the denominator equal to 0 are ecluded values. You do not need to include ecluded values in your answers unless they are asked for. EXAMPLE 3 Simplifying Rational Epressions with Trinomials Simplify each rational epression, if possible. k + 1 A k 2-4k - 5 k + 1 (k + 1)(k - 5) k (k + 1) 1 (k - 5) k - 5 Factor the numerator and the denominator when possible. Divide out common factors. Simplify. B y 2-16 y 2-8y + 16 (y + 4)(y - 4) ( y - 4)(y - 4) (y + 4)(y - 4) 1 (y - 4)(y - 4) 1 y + 4 _ y Simplifying Rational Epressions 867

23 Simplify each rational epression, if possible. r + 2 3a. 3b. b 2-25 r 2 + 7r + 10 b b + 25 Recall from Chapter 8 that opposite binomials can help you factor polynomials. Recognizing opposite binomials can also help you simplify rational epressions. Consider - 3. The numerator and denominator are opposite binomials. 3 - Therefore, _ = _ = ( - 3) = 1-1 =-1. EXAMPLE 4 Simplifying Rational Epressions Using Opposite Binomials Simplify each rational epression, if possible A _ 25-2 B 2( - 5) (5 - )(5 + ) Factor. 2( - 5) (5 - )(5 + ) Identify opposite binomials. 2( - 5) -1( - 5)(5 + ) 2( - 5) -1( - 5)(5 + ) - 2_ 5 + Rewrite one opposite binomial. Divide out common factors. Simplify. 2-2m 2m 2 + 2m - 4 2(1 - m) 2(m + 2)(m - 1) 2(1 - m) 2(m + 2)(m - 1) _ 2(1 - m) 2(m + 2)(-1)(1 - m) _ 2 1 (1 - m) (m + 2)(-1)(1 - m) 1 - m + 2 Simplify each rational epression, if possible. 4a _ b c _ tude Simplifying Rational Epressions When I can t tell if I m allowed to divide out part of an epression, I substitute a number into the original epression and simplify it. Then I simplify the original epression by dividing out the term in question. I check by seeing if the results are the same. For eample, I ll use = 2 to see if I can divide out 4 in _ Substitute = 2. Divide out 4. Tanika Brown, Washington High School 4_ 4-7 _ 4(2) _ (2) - 7 8_ 8-7 = _ 8 1 = 8 _ cannot be divided out because 8 _ Chapter 12 Rational Functions and Equations

24 EXAMPLE 5 Biology Application Desert plants must conserve water. Water evaporates from the surface of a plant. The volume determines how much water is in a plant. So the greater the surface-area-to-volume ratio, the less likely a plant is to survive in the desert. A barrel cactus is a desert plant that is close to spherical in shape. a. What is the surface-area-to-volume ratio of a spherical barrel cactus? (Hint: For a sphere, S = 4π r 2 and V = 4 3 π r 3 ) 4π r 2 _ 4 3 π r 3 Write the ratio of surface area to volume. 4π 1 r 2 _ 4 3 π 1 r 3 4r 2 1 _ 4 3 r 31 4_ r _ 3 4 Divide out common factors. Use properties of eponents. Multiply by the reciprocal of _ r 3 _ 4 1 3_ r Divide out common factors. Simplify. For two fractions with the same numerator, the value of the fraction with a greater denominator is less than the value of the other fraction. 9 > 3 2_ 9 < _ 2 3 b. Which barrel cactus has a greater chance of survival in the desert, one with a radius of 4 inches or one with a radius of 7 inches? Eplain. 3_ r = 3 _ 4 3_ r = 3_ 7 3_ 4 > 3 _ 7 Write the ratio of surface area to volume twice. Substitute 4 and 7 for r. Compare the ratios. The barrel cactus with a radius of 7 inches has a greater chance of survival. Its surface-area-to-volume ratio is less than for a cactus with a radius of 4 inches. 5. Which barrel cactus has less of a chance to survive in the desert, one with a radius of 6 inches or one with a radius of 3 inches? Eplain. THINK AND DISCUSS 1. Write a rational epression that has an ecluded value that cannot be identified when the epression is in its simplified form. 2. GET ORGANIZED Copy and complete the graphic organizer. In each bo, write and simplify one of the given rational epressions using the - 3 most appropriate method., 4 5, , Simplifying Rational Epressions 869

25 12-3 Eercises KEYWORD: MA GUIDED PRACTICE 1. Vocabulary What is true about both the numerator and denominator of rational epressions? KEYWORD: MA7 Parent SEE EXAMPLE 1 p. 866 Find any ecluded values of each rational epression. 2. 5_ m _ p 2 p 2-2p - 15 SEE EXAMPLE 2 p. 867 Simplify each rational epression, if possible. Identify any ecluded values. 5. _ 4a 2 8a 8. 10_ 5 - y 6. _ 2d d d h_ 2h _ y + 3 3( + 4) _ 6 SEE EXAMPLE 3 p. 867 Simplify each rational epression, if possible. 11. b + 4 b 2 + 5b s 2-4 _ s 2 + 4s c 2 + 5c + 6 (c + 3)(c - 4) 14. ( - 2)( + 1) j 2-25 j 2 + 2j p + 1 p 2-4p - 5 SEE EXAMPLE 4 p n - 16 _ 64 - n _ q - 50 _ q r r 2-4r a a 2 + 2a - 15 SEE EXAMPLE 5 p Construction The side of a triangular roof will have the same height h and base b 2 as the side of a trapezoidal roof. a. What is the ratio of the area of the triangular roof to the area of the trapezoidal roof? (Hint: For a triangle, A = 1 2 b 2h. For a trapezoid, A = b 1 + b 2 h.) 2 b. Compare the ratio from part a to what the ratio will be if b 1 is doubled for the trapezoidal roof and b 2 is doubled for both roofs? PRACTICE AND PROBLEM SOLVING Find any ecluded values of each rational epression. 24. c_ c 2 + c 25. 2_ n 2-1 2n 2-7n - 4 Simplify each rational epression, if possible. Identify any ecluded values. 28. _ 4d d 2 d _ 3m _ 10y 4 m - 4 2y 31. _ 2t 2 16t 870 Chapter 12 Rational Functions and Equations

26 Independent Practice For See Eercises Eample Etra Practice Skills Practice p. S26 Application Practice p. S39 Biology Simplify each rational epression, if possible. 32. q - 6 q 2-9q p 2-6p - 7 p 2-4p _ z 2-2z + 1 z b 3b b Geometry When choosing package sizes, a company wants a package that uses the least amount of material to hold the greatest volume of product. a. What is the surface-area-to-volume ratio for a rectangular prism? (Hint: For a rectangular prism, S = 2lw + 2lh + 2wh and V = lwh.) b. Which bo should the company choose? Eplain t - 3 _ t 2-5t v - 36 _ v Biology The table gives information on two populations of animals that were released into the wild. Suppose 16 more predators and 20 more prey are released into the area. Write and simplify a rational epression to show the ratio of predator to prey. Original Population Population 5 Years Later Predator 4 Prey 5 As a result of a nationwide policy of protection and reintroduction, the population of bald eagles in the lower 48 states grew from 417 nesting pairs in 1963 to more than 6400 nesting pairs in Source: U.S. Fish and Wildlife Service Simplify each rational epression, if possible. 43. p p p j - 5 _ j 2-25 ( + 1) n n n 3n n 47. 6w w - 7 6w _ ( + 5) a_ 2a + a n 2 - n - 56 n 2-16n This problem will prepare you for the Multi-Step Test Prep on page 876. It takes 250 workdays to build a house. The number of construction days is determined by the size of the crew. The crew includes one manager who supervises workers and checks for problems, but does not do any building. a. The table shows the number of construction days as a function of the number of workers. Copy and complete the table. b. Use the table to write a function that represents the number of construction days. c. Identify the ecluded values of the function. Crew Size () Workdays Workers 2 _ _ Construction Days (y) Simplifying Rational Epressions 871

27 53. Geometry Let s represent the length of an edge of a cube. a. Write the ratio of a cube s surface area to volume in simplified form. (Hint: For a cube, S = 6 s 2.) b. What is the ratio of the cube s surface area to volume when s = 2? c. What is the ratio of the cube s surface area to volume when s = 6? 54. Write About It Eplain how to find ecluded values for a rational epression. 55. Critical Thinking Give an eample of a rational epression that has in both the numerator and denominator, but cannot be simplified. 56. Which epression is undefined for = 4 and =-1? - 1 _ _ Which epression is the ratio of the area of a triangle to the area of a rectangle that has the same base and height? bh (bh) Gridded Response What is the ecluded value for ? CHALLENGE AND EXTEND Tell whether each statement is sometimes, always, or never true. Eplain. 59. A rational epression has an ecluded value. 60. A rational epression has a square root in the numerator. 61. The graph of a rational function has at least one asymptote. Simplify each rational epression v - 6 v 2 4v 2-4v a 2-7a + 3 2a 2 + 9a - 5 Identify any ecluded values of each rational epression y y _ SPIRAL REVIEW 68. A rectangle has an area of 24 square feet. Every dimension is multiplied by a scale factor, and the new rectangle has an area of 864 square feet. What is the scale factor? (Lesson 2-7) Use intercepts to graph the line described by each equation. (Lesson 5-2) y = y = _ y = 2 For each of the following, let y vary inversely as. (Lesson 12-1) = 2, y 1 = 4, and 2 = 1. Find y = 2, y 1 =-1, and y 2 = _ 1 3. Find Chapter 12 Rational Functions and Equations

28 12-3 Graph Rational Functions You can use a graphing calculator to graph rational functions and to compare graphs of rational functions before and after they are simplified. Use with Lesson 12-3 Activity _ - 1 Simplify y = and give any ecluded values. Then graph both the original function and the simplified function and compare the graphs. 1 Simplify the function and find the ecluded values = - 1 ( - 1)( - 4) = _ 1 ; ecluded values: 4, Enter y = - 1 and y = 1 into your calculator as shown and press. KEYWORD: MA7 Lab12 3 To compare the graphs, press. At the top of the screen, you can see which graph the cursor is on. To change between graphs, press and. 4 The graphs appear to be the same, but check the ecluded values, 4 and 1. While on Y1, press 4. Notice that there is no y-value at = 4. The function is undefined. 5 Press to switch to Y2 and press 4. This function is also undefined at = 4. The graphs are the same at this ecluded value. 6 Return to Y1 and press 1. This function is undefined at = 1. However, this is not a vertical asymptote. Instead, this graph has a hole at = 1. 7 Switch to Y2 and press 1. This function is defined at = 1. So the two graphs are the same ecept at = 1. Try This 1. Why is = 1 an ecluded value for one function but not for the other? 2. Are the functions y = - 1 and y = 1 truly equivalent for all values of? Eplain. 3. Make a Conjecture Complete each statment. a. If a value of is ecluded from a function and its simplified form, it appears on the graph as a(n)?. b. If a value of is ecluded from a function but not its simplified form, it appears on the graph as a? Technology Lab 873

29 Representing Solid Figures Geometry A net is a flat pattern that can be folded to make a solid figure. The net shows all of the faces and surfaces of the solid. See Skills Bank page S65 To identify the solid shown by a net, remember these properties of solids. Prisms Pyramids Cylinder Cone A prism has two parallel, congruent bases that are polygons. The other faces are rectangles or parallelograms. A base of a pyramid is a polygon. The other faces are triangles. A cylinder has two congruent circles as its bases. A cone has one circle as its base. Eample 1 Identify the solid shown by this net. The faces are all polygons, so this is either a prism or a pyramid. Look for a pair of congruent polygons. There are two congruent right triangles. The solid is a prism. A prism is named by the shape of its bases, so this is a triangular prism. Try This Identify the solid shown by each net Chapter 12 Rational Functions and Equations

30 Identify the solid shown by each net A foundation plan is a drawing that represents a solid made from cubes. The squares in the foundation plan are a top view of the solid. The number in each square shows the height of the solid at that point. The foundation plan shown is like a set of instructions for building the solid net to it. Eample 2 Draw a foundation plan for this solid figure. Use squares to show a top view of the solid. Put a number in each square to show the height at that point. Try This Draw a foundation plan for each solid figure Connecting Algebra to Geometry 875

31 SECTION 12A Rational Functions and Epressions Construction Daze Robert is part of a volunteer crew constructing houses for low-income families. The table shows how many construction days it takes to complete a house for work crews of various sizes. Crew Size Construction Days Workdays Working at the same rate, how many construction days should it take a crew of 40 people to build the house? 2. Epress the number of construction days as a function of the crew size. Define the variables. What type of relationship is formed in the situation? 3. Eplain how the crew size affects the number of construction days. 4. About how many construction days would it take a crew of 32 to complete a house? 5. If a crew can complete a house in 12.5 days, how many people are in the crew? 6. What are a reasonable domain and range of the function? 7. Suppose there are two managers that do not contribute to the work of building the house, yet are counted as part of the crew. Epress the number of construction days as a function of the crew size. What are the asymptotes of this function? Graph the function. 876 Chapter 12 Rational Functions and Equations

32 Quiz for Lessons 12-1 Through 12-3 SECTION 12A 12-1 Inverse Variation Tell whether each relationship represents an inverse variation. Eplain y y y = 3 _ 4. y + = 3 _ 4 5. y =-2 6. y = _ 5 7. Write and graph the inverse variation in which y = 3 when = Write and graph the inverse variation in which y = 4 when = The number of calculators Mrs. Hopkins can buy for the classroom varies inversely as the cost of each calculator. She can buy 24 calculators that cost $60 each. How many calculators can she buy if they cost $80 each? 12-2 Rational Functions Identify the ecluded value and the vertical and horizontal asymptotes for each rational function. Then graph each function. 10. y = 12 _ 11. y = 6_ y = 4_ y = 14. Jeff builds model train layouts. He has $75 to spend on packages of miniature landscape items. He receives 6 free packages with each order. The number of packages y that Jeff can buy is given by y = , where represents the cost of each package in dollars. Describe the reasonable domain and range values and graph the function. 2_ Simplifying Rational Epressions Find any ecluded values of each rational epression. 15. _ 15 n 16. p_ p t - 1 _ t 2 + t Simplify each rational epression, if possible. Identify any ecluded values. 19. _ n_ 21. s n 2-3n s 2-4s Suppose a cone and a cylinder have the same radius and that the slant height l of the cone is the same as the height h of the cylinder. Find the ratio of the cone s surface area to the cylinder s surface area Ready to Go On? 877

33 12-4 Multiplying and Dividing Rational Epressions Objective Multiply and divide rational epressions. Why learn this? You can multiply rational epressions to determine the probabilities of winning prizes at carnivals. (See Eample 5.) The rules for multiplying rational epressions are the same as the rules for multiplying fractions. You multiply the numerators, and you multiply the denominators. Multiplying Rational Epressions If a, b, c, and d are nonzero polynomials, then _ a b c_ d = _ ac bd. EXAMPLE 1 Multiplying Rational Epressions See the Quotient of Powers Property in Lesson 7-4. _ a m a n = am-n Multiply. Simplify your answer. a + 3 A _ 6_ 2 3a + 9 _ 6(a + 3) Multiply the numerators and denominators. 2(3a + 9) 6(a + 3) Factor. 2 3(a + 3) 6 1 _(a + 3) 1 Divide out the common factors. 6 1 (a + 3) 1 1 Simplify. 12b B _ 3 c 2 _ 15a 2 b 5ac 3b 2 c C (12)(15)a 2 (b 3 b)c 2 (5)(3)ab 2 (c c) _ 180a 2 b 4 c 2 15ab 2 c 2 12a 1 b 2 c 0 12ab 2 _ 5 2 _ 2y 3 3 2y 2 _ y 5 Multiply the numerators and the denominators. Arrange the epression so like variables are together. Simplify. Divide out common factors. Use properties of eponents. Simplify. Remember that c 0 = 1. Multiply. There are no common factors, so the product cannot be simplified. 878 Chapter 12 Rational Functions and Equations

34 Multiply. Simplify your answer. (c - 4) 1a. _ 45 5 (-4c + 16) 1b. _ 5y 5 z 3 y 2 z _ y 4y EXAMPLE 2 Multiplying a Rational Epression by a Polynomial Just as you can write an integer as a fraction, you can write any epression as a rational epression by writing it with a denominator of 1. _ Multiply ( ) _ ( + 3)( + 5) 4_ 1 2( + 3) ( + 3) 1 ( + 5) ( + 3) Multiply. Simplify your answer. Write the polynomial over 1. Factor the numerator and denominator. Divide out common factors. Multiply remaining factors. m - 5 3m + 6. Simplify your answer. m 2-4m - 12 There are two methods for simplifying rational epressions. You can simplify first by dividing out common factors and then multiply the remaining factors. You can also multiply first and then simplify. Using either method will result in the same answer. EXAMPLE 3 Multiplying Rational Epressions Containing Polynomials _ Multiply 4d 3 + 4d 16f _ 2f 7d 2 f + 7f. Simplify your answer. Method 1 Simplify first. _ 4 d 3 + 4d _ 2f 16f 7d 2 f + 7f 4d(d 2 + 1) 16f 4 1 d (d 2 + 1) f 1 Then multiply. d_ 14f 2f _ 7f (d 2 + 1) 2 1 f 1 7f (d 2 + 1) 1 Factor. Divide out common factors. Simplify. Method 2 Multiply first. _ 4d 3 + 4d _ 2f 16f 7d 2 f + 7f (4d 3 + 4d)2f 16f (7d 2 f + 7f ) 8d 3 f + 8df 112d 2 f f 2 Then simplify. 8df (d 2 + 1) 112f 2 (d 2 + 1) 8 1 d f 1 (d 2 + 1) f 21 (d 2 + 1) 1 Multiply Distribute. Factor. Divide out common factors. d_ 14 f Simplify. Multiply. Simplify your answer. 3a. _ n - 5 n 2 + 4n n 2 + 8n + 16 n 2-3n - 10 p + 4 3b. _ p 2 + 2p p 2-3p - 10 p Multiplying and Dividing Rational Epressions 879

35 The rules for dividing rational epressions are the same as the rules for dividing fractions. To divide by a rational epression, multiply by its reciprocal. Dividing Rational Epressions If a, b, c, and d are nonzero polynomials, then a _ b c _ d = a _ b d _ c = ad _ bc. EXAMPLE 4 Dividing by Rational Epressions and Polynomials Divide. Simplify your answer. A _ _ - 2 _ 1(2) ( - 2) B C _ ( - 2) 2_ - 2 _ _ _ ( - 2) ( + 1)( + 1) 2 - _ ( - 2) ( + 1)( + 1) -1( - 2) _ 1 ( - 2) 1 ( + 1)( + 1) 1-1( - 2) 1 -( + 1) 2 _ 3a 2 b (3a 2 + 6a) b _ 3a 2 b _ b 3a 2 + 6a 1 _ 3a 2 b b 3a 2 + 6a 3a 2 b b(3a 2 + 6a) 3 1 a 21 b 1 b a 1 (a + 2) a_ (a + 2) Write as multiplication by the reciprocal. Multiply the numerators and the denominators. Divide out common factors. Simplify. Write as multiplication by the reciprocal. Factor. Rewrite one opposite binomial. Divide out common factors. Multiply. Write the binomial over 1. Write as multiplication by the reciprocal. Multiply the numerators and the denominators. Factor. Divide out common factors. Simplify. Divide. Simplify your answer. 4a. 3_ 3 _ 2 ( - 5) 4c. 2 - _ + 2 ( ) 4b. 18vw 2 _ 6v _ 3v 2 4 2w Chapter 12 Rational Functions and Equations

36 EXAMPLE 5 Probability Application Marty is playing a carnival game. He needs to pick two items out of a bag without looking. The bag has red and blue items. There are three more red items than blue items. a. Write and simplify an epression that represents the probability that Marty will pick two blue items without replacing the first item. Let = the number of blue items. Blue + Red = Total Write epressions for the number of each color item and for the total = number of items. The probability of picking a blue item and then another blue item is the product of the probabilities of the individual events. 1st pick: blue items 2nd pick: blue items For a review of probability topics, see Chapter 10. 1st pick: total items = _ (2 + 3) _ - 1 2( + 1) ( - 1) = 2(2 + 3)( + 1) 2nd pick: total items b. What is the probability that Marty picks two blue items if there are 10 blue items in the bag before his first pick? Use the probability of picking two blue items. Since represents the number of blue items, substitute 10 for. 10(10-1) P (blue, blue) = Substitute. 2( )(10 + 1) 10(9) = _ 2(23)(11) = _ Use the order of operations to simplify. The probability of picking two blue items if there are 10 blue items in the bag before the first pick is approimately What if? There are 50 blue items in the bag before Marty s first pick. What is the probability that Marty picks two blue items? THINK AND DISCUSS 1. Eplain how to divide by a polynomial. 2. GET ORGANIZED Copy and complete the graphic organizer. In each bo, describe how to perform the operation with rational epressions Multiplying and Dividing Rational Epressions 881

37 12-4 Eercises KEYWORD: MA SEE EXAMPLE 1 p. 878 SEE EXAMPLE 2 p. 879 SEE EXAMPLE 3 p. 879 SEE EXAMPLE 4 p. 880 SEE EXAMPLE 5 p. 881 GUIDED PRACTICE Multiply. Simplify your answer. 1. 4hj 2 _ 10j _ 3h 3 k 3 h 3 k 3 4. _ ab c _ 2a 2 3c _ _ 4y 2yz 2 _ 3. _ _ 7c 4 d 10c 5a_ 21c 3 d 4y + 8 (y 2-4) 8. _ _ 6 6. _ 12p 2 q _ 15p 4 q 3 5p 12q 6 (5 + 10) 9. 3m_ 2 6m + 18 (m 2-7m - 30) 4p_ 8p + 16 (p 2-5p - 14) 11. a 2 _ a (a a + 25) 12. _ a 2 + 6ab 5 + 3a b 3a 2 b + 5ab _ j - 1 j 2-4j + 3 _ j 2-5j + 6 2j r r + 14 r 2-16 _ 2r + 8 r + 1 Divide. Simplify your answer. 19. _ 3a 4 b 2a 2 c _ 12a 2 c 3 8c 4 -c _ 4c + 4 (c 2 - c - 2) _ p 3 + 4pq p _ 6q 3-8 2q _ 2m 3 + 2m m 2-2m _ 4m m - 1 _ y - 8 y 2-1 _ y + 2 y 2-49 KEYWORD: MA7 Parent ( 2-25) Probability While playing a game, Rachel pulls two tiles out of a bag without looking and without replacing the first tile. The bag has two colors of tiles black and white. There are 10 more white tiles than black tiles. a. Write and simplify an epression that represents the probability that Rachel will pick a black tile, then a white tile. b. What is the probability that Rachel pulls a black tile and then a white tile if there are 5 black tiles in the bag before her first pick? Independent Practice For See Eercises Eample Etra Practice Skills Practice p. S27 Application Practice p. S39 PRACTICE AND PROBLEM SOLVING Multiply. Simplify your answer. 23. _ p 6 q 2 7r _ -3p2 3 r _ 3r 2 t 6st 2r 2 s 3 2 _ t r 4 s 2 3_ 2a + 6 (a 2 + 4a + 3) 27. 4m 2-8m ( ) _ y + 5 _ y m 2 + 6m - 16 (m 2 + 7m - 8) 6n n n 2 + 9n + 8 _ n 2-1 2n + 6 3a 2 b 5a a 2 b 2a + 4b 31. _ t a 3 b + 6 a 2 b 2 5t _ t - 10 Divide. Simplify your answer. 32. _ 6j 2 k 5 _ 4j 3 k 3 5j 3j 33. _ a - 4 (8a - 2 a 2 ) 34. a _ Chapter 12 Rational Functions and Equations

38 35. Entertainment A carnival game board is covered completely in small balloons. You throw darts at the board and try to pop the balloons. a. Write and simplify an epression describing the probability that the net two balloons popped are red and then blue. (Hint: Write the probabilities as ratios of the areas of rectangles.) b. What is the probability that the net two balloons popped are red and then blue if = 3? 36. /ERROR ANALYSIS / Which is incorrect? Eplain the error. 37. Critical Thinking Which of the following epressions is NOT equivalent to the other three? Eplain why. a. _ _ 2-6 8y 2 b. 6 y 2 _ 3y 4 _ d. Multiply or divide. Simplify your answer. 38. _ 5p 3 p 2 q 2q 3 _ 40. p 2 c. _ 10 4 y 2 5 y 2 2 y _ 4 y y _ m 2-18m 12m m m m 2 + 4m + 3 _ _ 2-9 (4 2-36) m - 3 m 2-2m - 4 _ 6m - 66 m 2-4m 44. Write About It Eplain how to divide _ 12w 4 7 _ w w 3 4 m _ 3 4m. 45. This problem will prepare you for the Multi-Step Test Prep on page 906. The size of an image projected on a screen depends on how far the object is from the lens, the magnification of the lens, and the distance between the image and the lens. Magnification of a lens is M = I O = y where I is the height of the image, O is the height of the object, is the distance of the object from the lens, and y is the distance of the image from the lens. a. If an object 16 cm high is placed 15 cm from the lens, it forms an image 60 cm from the lens. What is the height of the image? b. Marie moves the same object to a distance of 20 cm from the lens. If the image is the same size as part a, what is the distance between the image and the lens? c. What is the magnification of the lens? 12-4 Multiplying and Dividing Rational Epressions 883

39 46. Which epression is equivalent to_ t (t _ + 4) 2 t _ Identify the product -_ 20 b 2 a 2 _ 3ab 15b. _ t + 4 3? -_ a -4b 2-4b 2 _ 4b 2 a -_ b 2 4a 48. Which of the following is equivalent to_ 2 + 5? _ _ _ _ Short Response Simplify ( ). Show your work CHALLENGE AND EXTEND Simplify _ _ _ _ ( - 1) A comple fraction is a fraction that contains one or more fractions in the numerator or the denominator. Simplify each comple fraction. a (Hint: Use the rule b_ c = _ a b _ c d.) d _ c + 5 _ 2 y _ c _ z _ a + 1 _ 3 c 2 + 6c + 5 _ 2 y 55. a 2 + 6a + 5 2a + 2 c a + 5 z SPIRAL REVIEW 56. Jillian s mother told her to preheat the oven to at least 325 F. When Jillian went into the kitchen, the oven was already set to 200 F. Write and solve an inequality to determine how many more degrees Jillian should increase the temperature. (Lesson 3-2) 57. Pierce has $30 to spend on a night out. He already spent $12 on dinner and $9 on a movie ticket. He will spend some money m on movie-theatre snacks. Write and solve an inequality that will show all the values of m that Pierce can spend on snacks. (Lesson 3-2) Simplify each radical. Then add or subtract, if possible. (Lesson 11-7) t + 5 3t + 12t Identify the ecluded value and asymptotes for each rational function. (Lesson 12-2) 61. y = 4_ 62. y = 63. y =- _ y =- 2_ 65. y = _ y = Chapter 12 Rational Functions and Equations

40 12-5 Adding and Subtracting Rational Epressions Objectives Add and subtract rational epressions with like denominators. Add and subtract rational epressions with unlike denominators. Who uses this? Kayakers can use rational epressions to figure out travel time for different river trips. (See Eample 5.) The rules for adding rational epressions are the same as the rules for adding fractions. If the denominators are the same, you add the numerators and keep the common denominator. 3_ 8 + _ 2 8 = _ = 5 _ 8 Adding Rational Epressions with Like Denominators If a, b, and c represent polynomials and c 0, then _ a c + _ b c = _ a + b c. EXAMPLE 1 Adding Rational Epressions with Like Denominators Add. Simplify your answer. A 3b_ 5b_ b + 2 b 2 _ 3b + 5b = _ 8b1 b 2 b 21 B C = _ 8 b _ 2-8 _ = ( - 2)( - 4)1 = _ 2m + 4 2_ m m 2-9 2m m 2-9 = _ 2m + 6 m 2-9 = ( - 2) Simplify. 2(m + 3) 1 = (m - 3)(m + 3) 1 = 2_ m - 3 Combine like terms in the numerator. Divide out common factors. Simplify. Combine like terms in the numerator. Factor. Divide out common factors. Combine like terms in the numerator. Factor. Divide out common factors. Simplify. Add. Simplify your answer. 1a. n_ 2n + _ 3n 2n 1b. _ 3y 2 y _ 3y y Adding and Subtracting Rational Epressions 885

41 To subtract rational epressions with like denominators, remember to add the opposite of each term in the second numerator. EXAMPLE 2 Subtracting Rational Epressions with Like Denominators Subtract. Simplify your answer. 3m - 6 m 2 + m m + 2 m 2 + m - 6 Make sure you add the opposite of all the terms in the numerator of the second epression when subtracting rational epressions. 3m (-m + 2) m 2 + m - 6 = 3m m - 2 m 2 + m - 6 = = = 4m - 8 m 2 + m - 6 Subtract numerators. Combine like terms. 4(m - 2) 1 Factor. Divide out common (m + 3)(m - 2) 1 factors. 4_ m + 3 Simplify. Subtract. Simplify your answer. 2a. _ 5a + 2 a _ 2a - 4 a 2-4 2b. 2b + 14 b 2 + 3b b + 2 b 2 + 3b - 4 As with fractions, rational epressions must have a common denominator before they can be added or subtracted. If they do not have a common denominator, you can use the least common multiple, or LCM, of the denominators to find one. To find the LCM, write the prime factorization of both epressions. Use each factor the greatest number of times it appears in either epression. 6 2 = = 5( + 3) 8 = = ( + 3) ( - 3) LCM = = 24 2 LCM = 5( + 3)( - 3) EXAMPLE 3 Identifying the Least Common Multiple Find the LCM of the given epressions. A 24 a 3, 4a 24a 3 = a a a 4a = 2 2 a LCM = a a a = 24 a 3 Write the prime factorization of each epression. Use every factor of both epressions the greatest number of times it appears in either epression. B 2 d d + 12, d 2 + 7d d d + 12 = 2 (d 2 + 5d + 6) Factor each epression. = 2(d + 3)(d + 2) Use every factor of both epressions d 2 + 7d + 12 = (d + 3) (d + 4) LCM = 2(d + 3)(d + 2)(d + 4) the greatest number of times it appears in either epression. Find the LCM of the given epressions. 3a. 5 f 2 h, 15f h 2 3b , ( - 6)( + 5) 886 Chapter 12 Rational Functions and Equations

42 The LCM of the denominators of fractions or rational epressions is also called the least common denominator, or LCD. You use the same method to add or subtract rational epressions. Step 1 Identify the LCD. Adding or Subtracting Rational Epressions Step 2 Multiply each epression by an appropriate form of 1 so that each term has the LCD as its denominator. Step 3 Write each epression using the LCD. Step 4 Add or subtract the numerators, combining like terms as needed. Step 5 Factor. Step 6 Simplify as needed. EXAMPLE 4 Adding and Subtracting with Unlike Denominators Epressions like m - 3 and 3 - m are opposite binominals. 3 - m = -1(m - 3) and m - 3 = -1(3 - m) Add or subtract. Simplify your answer. 3 A _ = 2 3 Step 1 4 = 2 2 Identify the LCD. LCD = = 12 2 B Step 2 Step 3 Step 4 Step 5 Step 6 5_ m m _ 3 ) _ 3 6 ( 2_ 2 2 ) + _ 2 4 ( 3 Multiply each epression by an appropriate form of 1. _ _ Write each epression using the LCD. _ Add the numerators (2 + ) Factor and divide out common factors. 2 + _ 2 Simplify. Step 1 The denominators are opposite binomials. The LCD can be either m - 3 or 3 - m. Identify the LCD. Step 2 m - 3-5_ 3 - m ( -1 Step 3 m _-5 m (-5) Step 4 _ m - 3 Steps 5, 6 6_ m ) Multiply the second epression by -1 to get an LCD of m Write each epression using the LCD. Subtract the numerators. Add or subtract. Simplify your answer. 4a. 4_ 3d - _ 2d 4b. a 2 + 4a 2d 3 a 2 + 2a _ a - 2 No factoring is needed, so just simplify Adding and Subtracting Rational Epressions 887

43 EXAMPLE 5 Recreation Application Katy wants to find out how long it will take to kayak 1 mile up a river and return to her starting point. Katy s average paddling rate is 4 times the speed of the river s current. a. Write and simplify an epression for the time it will take Katy to kayak the round-trip in terms of the rate of the river s current. Step 1 Write epressions for the distances and rates in the problem. The distance in both directions is 1 mile. Let represent the rate of the current, and let 4 represent Katy s paddling rate. Katy s rate against the current is 4 -, or 3. Katy s rate with the current is 4 +, or 5. Step 2 Use a table to write epressions for time. To write epressions for time in terms of distance and rate, solve d = rt for t. t = d _ r Direction Upstream (against current) Downstream (with current) Distance (mi) Rate (mi/h) Time (h) = _ distance rate _ _ 5 Step 3 Write and simplify an epression for the total time. total time = time upstream + time downstream total time = _ _ 1 Substitute known values. 5 = _ 1 3 ( 5_ 5 ) + _ 1 5 ( 3_ Multiply each fraction by an 3 ) appropriate form of 1. = 5_ _ Write each epression using 15 the LCD, 15. = 8_ Add the numerators. 15 b. If the rate of the river is 2 miles per hour, how long will it take Katy to kayak round trip? 8_ 15(2) = _ 4 Substitute 2 for. Simplify. 15 It will take Katy 4 of an hour, or 16 minutes, to kayak the round-trip What if? Katy s average paddling rate increases to 5 times the speed of the current. Now how long will it take Katy to kayak the round trip? THINK AND DISCUSS 1. Eplain how to find the least common denominator of rational epressions. 2. GET ORGANIZED Copy and complete the graphic organizer. In each bo, compare and contrast operations with fractions and rational epressions. 888 Chapter 12 Rational Functions and Equations

44 12-5 Eercises KEYWORD: MA KEYWORD: MA7 Parent GUIDED PRACTICE SEE EXAMPLE 1 p. 885 SEE EXAMPLE 2 p. 886 SEE EXAMPLE 3 p. 887 p _ - _ , ( + 5)( - 4) 9. y 2-16, (y + 9)(y - 4) Add or subtract. Simplify your answer. 3 -_ _ c 3c 5 7a - 2 5a a 2 + 3a + 2 a 2 + 3a + 2 Find the LCM of the given epressions. 7. 3y 2, 6 3yz SEE EXAMPLE _ +_ Subtract. Simplify your answer. 7 -_ 3 4. _ p. 886 SEE EXAMPLE Add. Simplify your answer. y 5y 4m + 30 m 2 + 8m _2 + _2 2. _ + m+5 m+5 3y 3y _ _ 12. _ Travel The Escobar family went on a car trip. They drove 100 miles on country roads and 240 miles on the highway. They drove 50% faster on the highway than on the country roads. Let r represent their rate on country roads in miles per hour. a. Write and simplify an epression that represents the number of hours it took the Escobar family to complete their trip in terms of r. (Hint: 50% faster means 150% of the original rate.) b. Find their total travel time if they drove the posted speed limit. PRACTICE AND PROBLEM SOLVING Independent Practice For See Eercises Eample Etra Practice Skills Practice p. S27 Application Practice p. S39 Add. Simplify your answer. 4y _ 4y a2-3 + _ 2a _ _ a + 3 a +3 y y Subtract. Simplify your answer. c+3 -c + 8 m2 - _ 6m 18. _ 17. _ -_ m-6 m-6 4c c a + 2-2a 2-9a - -5a 19. _ a-2 a-2 Find the LCM of the given epressions jk 4m, 25jm a 2 + 4a, 27a p 2-3p, pqr y 2z, 10y , y 2 + 7y + 10, y Add or subtract. Simplify your answer. y2 - y 2y _ _ 27. -_ y - 9 y - 4y z +_ _ 2 7z 3z _ 30. _ t + 4-3t - _ 28. _ t-4 4-t 3m - m2 31. _ 4m - 8 m 2-4m Adding and Subtracting Rational Epressions 889

45 Travel The first transcontinental railroad was completed in Utah on May 10, The occasion was commemorated with a golden spike that connected the eastern and western tracks. 32. Fitness Ira walks one mile from his house to the recreation center. After playing basketball, he walks home at only 85% of his normal walking speed. Let w be Ira s normal rate of walking. a. Write an epression to represent Ira s round-trip walking time. b. If Ira s normal rate of walking is 3 miles per hour, how long did it take for him to complete his walking? 33. Travel A train travels 500 miles across the Midwest 50 miles through cities and 450 miles through open country. As it passes through cities, it slows to one-fifth the speed it travels through open territory. Let r represent the rate in open territory in miles per hour. a. Write and simplify an epression that represents the number of hours it takes the train to travel 500 miles in terms of r. b. Find the total travel time if the train s rate through open territory is 50 miles per hour. c. Critical Thinking If you knew the time it took the train to make the roundtrip, how could you find its average rate? Add or subtract. Simplify your answer _ 5 + y + _ 2y y b_ 2b 3 + 3_ 3b 2 2y _ 8y 2 + 9_ 4y _ 49 - c 2 - c_ 49 - c _ r 2 + 2r r _ 2r + 9 r _ _ a_ a _ 12 - a 2-2 _ _ _ 2y 3y _ y + 1 y /ERROR ANALYSIS / Two students were asked to find the ecluded values of p the epression. Student A identified the ecluded value as - 4 p 2 - p - 12 p 2 - p - 12 p =-3. Student B identified the ecluded values as p =-3 and p = 4. Who is incorrect? What is the error? 44. Multi-Step At the spring fair there is a square Velcro target as shown. A player tosses a ball, which will stick to the target in some random spot. If the ball sticks to a spot in either the small square or the circle, the player wins a prize. What is the probability that a player will win a prize, assuming the ball sticks somewhere on the target? Round your answer to the nearest hundredth. 45. Critical Thinking Write two epressions whose sum is This problem will prepare you for the Multi-Step Test Prep on page 906. Jonathan is studying light in his science class. He finds that a magnifying glass can be used to project upside-down images on a piece of paper. The equation 1 f = y relates the focal length of the lens f, the distance of the object from the lens, and the distance of the image from the lens y. The focal length of Jonathan s lens is 12 cm. a. Jonathan wants to write y, the distance of the image from the lens, as a function of, the distance of the object from the lens. To begin, he rewrote the equation as 1 y = Eplain how he did this. b. Eplain how Jonathan simplified the equation in part a to 1 y = _ Chapter 12 Rational Functions and Equations

46 47. Critical Thinking Identify three common denominators that could be used to add _ 3 to Write About It Eplain how to find the least common denominator of two rational epressions when the denominators are opposite binomials. 49. What is the LCD of 6_ 3p + 3 and 4_ p + 1? p p + 1 3p Simplify_ _ 1. 3_ 5_ 3_ Which of the following is equivalent to_ 2-2? _ _ - 2 2_ _ 4-2 _ _ + 2 _ _ Etended Response Andrea biked 3 miles to the post office and 5 miles to the library. The rate at which she biked to the library was three times faster than her rate to the post office r. a. Write an epression that represents Andrea s total biking time in hours. Eplain what each part of your epression means in the situation. b. Simplify the epression. c. How long did it take Andrea to bike the 8 miles if her biking rate to the post office was 3 miles per hour? CHALLENGE AND EXTEND Add or subtract and simplify. Find the ecluded values _ + y - _ 2 + y y 2 3_ 2m + 4_ m + 2_ m a_ y + b _ z + c _ yz 1-1 y 56. Simplify the comple fraction. (Hint: Simplify the numerator and denominator of the comple fraction first.) y - 1 SPIRAL REVIEW Sketch a graph for each situation. (Lesson 4-1) 57. Snow falls lightly at first, then falls heavily at a steady rate. 58. Snow melts quickly during the afternoon, then stops melting at night. 59. Snow falls heavily, is shoveled away, and then a light snow falls. Solve each quadratic equation by factoring. (Lesson 9-6) 60. d 2-4d - 12 = g 2-9g = = 0 Simplify each rational epression, if possible. Identify any ecluded values. (Lesson 12-3) 63. _ 2t n 2 + 5n 65. _ 4 - t 2-4 n 2 + 3n Adding and Subtracting Rational Epressions 891

47 12-6 Model Polynomial Division Some polynomial divisions can be modeled by algebra tiles. If a polynomial can be modeled by a rectangle, then its factors are represented by the length and width of the rectangle. If one factor is a divisor, then the other factor is a quotient. Use with Lesson 12-6 KEY = 1 =-1 = =- = 2 Activity 1 Use algebra tiles to find the quotient ( ) ( + 2). Model Try to form a rectangle with a length of + 2. Place the 2 - tile in the upper-left corner. Then place two unit tiles in a row at the lower-right corner. Try to use all the remaining tiles to complete a rectangle. If you can complete a rectangle, then the width of the rectangle is the quotient. The rectangle has length + 2 and width + 3. So, ( ) ( + 2) = + 3. You can check ( + 3)( + 2) your answer Use the FOIL method. by multiplying Try This Use algebra tiles to find each quotient. 1. ( ) ( + 1) 2. ( ) ( + 5) 3. ( ) ( - 1) 4. ( ) ( + 2) 5. ( ) ( - 2) 6. ( ) ( + 1) 7. Describe what happens when you try to model ( ) ( + 1). 892 Chapter 12 Rational Functions and Equations

48 12-6 Dividing Polynomials Objective Divide a polynomial by a monomial or binomial. Why learn this? Division of polynomials can be used to compare the energy produced by solar panels. (See Eercise 50.) The electrical power (in watts) produced by a solar panel is directly proportional to the surface area of the solar panel. Division of polynomials can be used to compare energy production by solar panels of different sizes. To divide a polynomial by a monomial, you can first write the division as a rational epression. Then divide each term in the polynomial by the monomial. EXAMPLE 1 Dividing a Polynomial by a Monomial Divide ( ) _ _ _ _ _ _ Write as a rational epression. Divide each term in the polynomial by the monomial 2. Divide out common factors. Simplify. Divide. 1a. (8p 3-4 p p) (-4p 2 ) 1b. ( ) 6 Division of a polynomial by a binomial is similar to division of whole numbers. Dividing Polynomials WORDS NUMBERS POLYNOMIALS Step 1 Factor the numerator and/or denominator if possible. _ = _ r 2 + 3r + 2 (r = + 2)(r + 1) r + 2 (r + 2) Step 2 Divide out any common factors _ 3 (r + 2) (r + 1) (r + 2) Step 3 Simplify. 56 r Dividing Polynomials 893

49 EXAMPLE 2 Dividing a Polynomial by a Binomial Put each term of the numerator over the denominator only when the denominator is a monomial. If the denominator is a polynomial, try to factor first. Divide. A B c 2 + 4c - 5 c - 1 (c + 5)(c - 1) c - 1 (c + 5)(c - 1) 1 (c - 1) 1 c (3 + 2)( - 4) 4 - (3 + 2)( - 4) -1( - 4) (3 + 2)( - 4) 1-1( - 4) Factor the numerator. Divide out common factors. Simplify. Factor the numerator. Factor one opposite binomial. Divide out common factors. Simplify. Divide k + k2 2a. k + 2 2b. b 2-49 _ b + 7 2c. s s + 36 s + 6 Recall how you used long division to divide whole numbers 15 as shown at right. You can also use long division to divide polynomials. An eample is shown below Divisor Quotient Dividend Using Long Division to Divide a Polynomial by a Binomial Step 1 Write the binomial and polynomial in long division form. Step 2 Divide the first term of the dividend by the first term of the divisor. This is the first term of the quotient. Step 3 Multiply this first term of the quotient by the binomial divisor and place the product under the dividend, aligning like terms. Step 4 Subtract the product from the dividend. Step 5 Bring down the net term in the dividend. Step 6 Repeat Steps 2 5 as necessary until you get 0 or until the degree of the remainder is less than the degree of the binomial. 894 Chapter 12 Rational Functions and Equations

50 EXAMPLE 3 Polynomial Long Division Divide using long division. A ( ) ( + 2) Step Step Step Step ( 2 + 2) 0 + Write in long division form with epressions in standard form. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms. Subtract the product from the dividend. You can check your simplified answer by multiplying it by the divisor. When the remainder is 0, you should get the numerator. Step ( 2 + 2) Step ( 2 + 2) ( + 2) 0 Bring down the net term in the dividend. Repeat Steps 2 5 as necessary. The remainder is 0. Check Multiply the answer and the divisor. ( + 2)( + 1) B Write in long division form ( 2 + ) ( ) 0 2 = Multiply ( + 1). Subtract. Bring down the 3. 3 = 3 Multiply 3( + 1). Subtract. The remainder is 0. Check Multiply the answer and the divisor. ( + 1)( + 3) Divide using long division. 3a. (2y 2-5y - 3) (y - 3) 3b. (a 2-8a + 12) (a - 6) 12-6 Dividing Polynomials 895

51 Sometimes the divisor is not a factor of the dividend, so the remainder is not 0. Then the remainder can be written as a rational epression. EXAMPLE 4 Long Division with a Remainder Divide ( ) ( - 2) (2 2-4) 7-6 -(7-14) 8 Write in long division form. 2 2 = 2 Multiply 2( - 2). Subtract. Bring down the = 7 Multiply 7( - 2). Subtract. The remainder is _ - 2 8_ - 2 Write the remainder as a rational epression using the divisor as the denominator. Write the quotient with the remainder. Divide. 4a. (3m 2 + 4m - 2) (m + 3) 4b. (y 2 + 3y + 2) (y - 3) Sometimes you need to write a placeholder for a term using a zero coefficient. This is best seen if you write the polynomials in standard form. EXAMPLE 5 Dividing Polynomials That Have a Zero Coefficient Recall from Chapter 7 that a polynomial in one variable is written in standard form when the degrees of the terms go from greatest to least. Divide ( ) (2 + 3). ( ) (2 + 3) ( ) ( ) (-6-9) -6 Write the polynomials in standard form. Write in long division form. Use 0 2 as a placeholder for the 2 term = -2 2 Multiply -2 2 (2 + 3). Subtract. Bring down = 3 Multiply 3 (2 + 3). Subtract. Bring down = -3 Multiply -3(2 + 3). Subtract. The remainder is -6. ( ) (2 + 3) = _ Divide. 5a. ( ) ( - 2) 5b. (4p p 3 ) (p + 1) 896 Chapter 12 Rational Functions and Equations

52 THINK AND DISCUSS 1. When dividing a polynomial by a binomial, what does it mean when the remainder is 0? 2. Suppose that the final answer to a polynomial division problem is Find an ecluded value. Justify your answer GET ORGANIZED Copy and complete the graphic organizer. In each bo, show an eample Eercises KEYWORD: MA SEE EXAMPLE 1 p. 893 SEE EXAMPLE 2 p. 894 SEE EXAMPLE 3 p. 895 SEE EXAMPLE 4 p. 896 SEE EXAMPLE 5 p. 896 GUIDED PRACTICE Divide. 1. (4 2 - ) 2 2. (16a 4-4 a 3 ) 4a 3. (21b 2-14b + 24) 3b 4. (18r 2-12r + 6) -6r 5. ( ) (5m m 2-10) 5 m _ t2-6t t Divide using long division. 8. a2 - a y2 + 11y - 10 a - 4 3y p2 - p - 20 p (c 2 + 7c + 12) (c + 4) 14. (3s 2-12s - 15) (s - 5) (a 2 + 4a + 3) (a + 2) 18. (2r r + 5) (r - 3) 19. (n 2 + 8n + 15) (n + 4) 20. (2t 2 - t + 4) (t - 1) 21. (8n 2-6n - 7) (2n + 1) 22. (b 2 - b + 1) (b + 2) 23. ( ) (3 + ) 24. (3p 3-2 p 2-4) (p - 2) 25. (m 2 + 2) (m - 1) 26. ( ) (5 + ) 27. (4k 3-2k - 8) (k + 1) 28. (j 3 + 6j + 2) (j + 4) KEYWORD: MA7 Parent PRACTICE AND PROBLEM SOLVING Divide. 29. (9t t 2-6t) 3 t (5n 3-10n + 15) -5n 31. (-16p p 3 + 8) 4 p r2-9r + 2 r t 2 + 2t - 3 2t g 2 + 7g - 6 g Dividing Polynomials 897

53 Independent Practice For See Eercises Eample Etra Practice Skills Practice p. S27 Application Practice p. S39 Solar Energy Divide using long division. 35. ( ) ( - 2) 36. (2m 2 + 8m + 8) (m + 2) 37. (6a 2 + 7a - 3) (2a + 3) 38. ( ) ( - 4) 39. ( ) ( - 2) 40. (2m 2 + 5m + 8) (m + 1) 41. ( ) (2-1) 42. (2m 3-4m - 30) (2m - 10) 43. (6t t + 9) (3t + 9) 44. (p 4-7 p 2 + p + 1) (p - 3) 45. Multi-Step Find the value of n, so that - 4 is a factor of n. Geometry The area of each of three rectangles is c m 2. Below are the different widths of the rectangles. Find each corresponding length Graphing Calculator Use the table of values for f () = ( ) to answer - 5 the following. a. Describe what is happening to the values of y as increases from 2 to 4. b. Describe what is happening to the values of y as increases from 6 to 8. c. Eplain why there is no value in the y column when is Estimation Estimate the value of for = NASA is researching unmanned solarpowered aircraft. In the future, these aircraft may be able to stay in the air indefinitely. 51. Solar Energy The greater the area of a solar panel, the greater the number of watts of energy produced. The area of two solar panels A and B, in square meters, can be represented by A = m 2 + 3m + 2 and B = 2m + 2. Divide the polynomials to find an epression that represents the ratio of the area of A to the area of B. 52. /ERROR ANALYSIS / Two students attempted to divide Which is incorrect? Eplain the error. 53. This problem will prepare you for the Multi-Step Test Prep on page 906. Jonathan continues to study lenses and uses the equation 1 y = _ a. Jonathan wants to write y, the distance of the image from the lens, as a function of, the distance of the object from the lens. What is the equation solved for y? b. Use a graphing calculator to create a table of values for the function y(). For which value of is the function undefined? 898 Chapter 12 Rational Functions and Equations

54 54. Write About It When dividing a polynomial by a binomial, what does it mean when there is a remainder? 55. Critical Thinking Divide Find a value for each epression by substituting 10 for in the original problem. Repeat the division. Compare the results of each division. 56. Write About It Is a factor of ? Eplain. 57. Which epression has an ecluded value of -_ 1 2? Find ( 2-1) ( + 2) _ _ Which epression is a factor of ? _ _ Which of the following epressions is equivalent to ( ) ( - 1)? _ _ _ _ 3-1 CHALLENGE AND EXTEND Divide. Simplify your answer. 61. (6 3 y y 2 ) (2 2 y) 62. ( 3-1) ( - 1) 63. ( ) ( 2-1) 64. ( 3 + 8) ( + 2) 65. Geometry The base of a triangle is m and the area is m 2. How much longer is the base than the height? 66. Geometry The formula for finding the volume of a cylinder is V = BH, where B is the area of the base of the cylinder and H is the height. a. Find the height of the cylinder given that V = π ( ) and B = π ( ). b. Find an epression for the radius of the base. SPIRAL REVIEW 67. Find the probability that a point within the boundaries of the larger rectangle is in the shaded region. Epress your answer as a simplified radical epression. (Lesson 11-8) Multiply. Write each product in simplest form. (Lesson 11-8) (6-10) 70. ( 3 + 2)( 3 + 5) Multiply. Simplify your answer. (Lesson 12-4) _ 72. _ 9 y 2( + 3) 2 3 _ 8y k k 3 _ k + 1 k2 + 3k + 2 2k Dividing Polynomials 899

55 12-7 Solving Rational Equations Objectives Solve rational equations. Identify etraneous solutions. Vocabulary rational equation Why learn this? Rational equations can be used to find how much time it takes two or more people working together to complete a job. (See Eample 3.) A rational equation is an equation that contains one or more rational epressions. If a rational equation is a proportion, it can be solved using the Cross Product Property. EXAMPLE 1 Solving Rational Equations by Using Cross Products 3_ 2_ Solve t - 3 =. Check your answer. t 3_ t - 3 = _ 2 Use cross products. t 3t = (t - 3)(2) 3t = 2t - 6 Distribute 2 on the right side. t =-6 Subtract 2t from both sides. Check 3_ t - 3 = _ 2 t 3_ 2_ _ -9-1 _ 3-6 2_ -6-1 _ 3 Solve. Check your answer. 1a. _ 1 n = 3_ 1b. 4_ n + 4 h + 1 = _ 2 h 1c. 2-7 = 3 _ Some rational equations contain sums or differences of rational epressions. To solve these, you must find the LCD of all the rational epressions in the equation. EXAMPLE 2 Solving Rational Equations by Using the LCD 3_ Solve 2_ c + 2c = c + 1. Step 1 Find the LCD. 2c(c + 1) Include every factor of the denominators. Step 2 Multiply both sides by the LCD. 2c(c + 1) ( c + _ 2c) 3 = 2c(c + 1) 2_ ( c + 1) c + 1) 2c(c + 1)( 1 _ c ) + 2c(c + 1) ( 3_ 2c ) = 2c(c + 1) ( 2_ Distribute on the left side. 900 Chapter 12 Rational Functions and Equations

56 Step 3 Simplify and solve. 1) Divide out common c + 1 factors. 2(c + 1) + (c + 1)3 = (2c)2 Simplify. 2c c + 3 = 4c Distribute and multiply. 5c + 5 = 4c Combine like terms. c + 5 = 0 Subtract 4c from both sides. c =-5 Subtract 5 from both sides. 2c 1 (c + 1) ( c 1) + 2c1 (c + 1) ( 3_ 2c 1) = 2c(c + 1)1 ( 2_ Check Verify that your solution is not etraneous. c + _ 3 2c = 2_ c + 1 2_ -5 + _ 3 2(-5) 2_ _ _ 10-1 _ _ -4-1 _ 2-1 _ 2 Solve each equation. Check your answer. 2a. 2_ a a + 1 = _ 4 a 2b. 6_ j _ 10 j 2c. 8_ t + 3 = 1 _ t + 1 = 4 _ 2j EXAMPLE 3 Problem-Solving Application Greg can clean a house in 5 hours. It takes Armin 7 hours to clean the same house. How long will it take them to clean the house if they work together? 1 Understand the Problem The answer will be the number of hours h Greg and Armin need to Armin 7 hours clean the house. List the important information: Greg cleans the house in 5 hours, so he cleans 1 5 Armin cleans the house in 7 hours, so he cleans Make a Plan Greg 5 hours of the house per hour. of the house per hour. The part of the house Greg cleans plus the part of the house Armin cleans equals the complete job. Greg s rate times the number of hours worked plus Armin s rate times the number of hours worked will give the complete time to clean the house. Let h represent the number of hours worked. (Greg s rate) h + (Armin s rate) h = complete job 5 h + 7 h = Solving Rational Equations 901

57 3 Solve 35 ( 5 h + 1 _ 7 h ) = 35(1) Multiply both sides by the LCD, 35. 7h + 5h = 35 12h = 35 h = _ = 2 _ Distribute 35 on the left side. Combine like terms. Divide by 12 on both sides. Greg and Armin working together can clean the house in 2 11 hours, or 12 2 hours 55 minutes. 4 Look Back Greg cleans 1 5 of the house per hour and Armin cleans 1 of the house per 7 hour. So, in 2 11 hours, Greg cleans = 7 of the house and Armin 12 cleans = 5 of the house. Together, they clean = 1 house. 3. Cindy mows a lawn in 50 minutes. It takes Sara 40 minutes to mow the same lawn. How long will it take them to mow the lawn if they work together? When you multiply each side of an equation by the LCD, you may get an etraneous solution. Recall from Chapter 11 that an etraneous solution is a solution to a resulting equation that is not a solution to the original equation. EXAMPLE 4 Etraneous Solutions _ Solve - 9 _ 2-9 = -3. Identify any etraneous solutions. - 3 Etraneous solutions may be introduced by squaring both sides of an equation or by multiplying both sides of an equation by a variable epression. Step 1 Solve. ( - 9)( - 3) =-3( 2-9) Use cross products. Multiply the left side. Distribute -3 on the right side. Add 3 2 to both sides = = = 0 4( - 3) = 0 4 = 0 or - 3 = 0 = 0 or = 3 Subtract 27 from both sides. Factor the quadratic epression. Use the Zero Product Property. Solve for. Step 2 Find etraneous solutions. _ = _-3 _ = _-3-3 _ 0-9 _-3 _ 3-9 _-3 Because both _-9 _-3 _-6 _ The only solution is 0, so 3 is an etraneous solution. -3 and _ 0 are undefined, 3 is not a solution. 902 Chapter 12 Rational Functions and Equations Solve. Identify any etraneous solutions. 4a. 3_ - 7 = _ - 2 4b. _ = 4_ - 3 4c. 9_ = _ 6 2

58 THINK AND DISCUSS 1. Why is it important to check your answers to rational equations? 2. For what values of are the rational epressions in the equation - 3 = undefined? 3. Eplain why some rational equations, such as - 4 = 4, have no - 4 solutions. 4. GET ORGANIZED Copy and complete the graphic organizer. In each bo, write the solution and check Eercises KEYWORD: MA GUIDED PRACTICE KEYWORD: MA7 Parent 1. Vocabulary A(n)? contains one or more rational epressions. (etraneous solution or rational equation) SEE EXAMPLE 1 p. 900 SEE EXAMPLE 2 p. 900 SEE EXAMPLE 3 p. 901 SEE EXAMPLE 4 p. 902 Solve. Check your answer _ + 4 = 2 _ _ j = j _ - 5 _ = 1 _ _ 8 d = d _ 3 d _ a - 4 = a_ a _ s - 6 = 4 _ s 4. 3_ - 4 = 9_ a _ _ 3 = 2 _ _ s - 6 = 4 _ s + 1 _ 2s r_ 2-2 _ r = 5 _ 6 20_ p = _ -10 2p 6_ 2-1 = 3 3_ + 1 = 2 _ + 3 _ 13. _ 7 r + 2_ r - 1 = _ -1 2r _ n = 7 _ n 2-1 2_ + 1 = _ a = _ -4 2 a _ 1 p = _ -3 p Painting Summer can paint a room in 3 hours. Louise can paint the room in 5 hours. How many hours will it take to paint the room if they work together? 21. Technology Lawrence s old robotic vacuum can clean his apartment in 1 1 hours. His new robotic vacuum can clean his apartment in 45 minutes. 2 How long will it take both vacuums, working together, to clean his apartment? Solve. Identify any etraneous solutions _ c - 4 = c - 1 _ c w + 3 _ w w_ = _ _ w = 8_ Solving Rational Equations 903

59 Independent Practice For See Eercises Eample Etra Practice Skills Practice p. S27 Application Practice p. S39 PRACTICE AND PROBLEM SOLVING Solve. Check your answer _ - 2 = 2_ _ s - 2 _ s = _ 3n - 1 = _ 3 n _ 1 = _ r_ 3-3 = - _ 6 r 35. _ + 4 = _ c - 2 _ c = _ - 1 4_ c _ + 5 = _ 4 9_ m - 3_ 2m = _ 15 m 8_ 3 = _ _ _ = _ _ = _ _ Mel can carpet a floor in 10 hours. Sandy can carpet the same floor in 15 hours. How many hours will it take them to carpet the floor if they work together? Solve. Identify any etraneous solutions. 38. _ 5-3 = _ _ 2 = _ _ 3t t - 3 = _ t + 4 t _ = - 4 _ Multi-Step Clancy has been keeping his free throw Clancy s Free Throws statistics. Use his data to write the ratio of the number of free throws Clancy has made to the number of attempts. Attempts Made a. What percentage has he made? b. Write and solve an equation to find how many free throws f Clancy would have to make in a row to improve his free-throw percentage to 90%. (Hint: Clancy needs to make f more free throws in f more attempts.) 43. Travel A passenger train travels 20 mi/h faster than a freight train. It took the passenger train 2 hours less time than the freight train to travel 240 miles. The freight train took t hours. Copy and complete the chart. Then find the rate of the freight train. Distance Rate Time Passenger train t - 2 Freight train 240 t 44. Pipe A fills a storage tank with a certain chemical in 12 hours. Pipe B fills the tank in 18 hours. How long would it take both pipes to fill the tank? CLOSE TO HOME 1997 John McPherson. Reprinted with permission of UNIVERSAL PRESS SYNDICATE. All rights reserved. 45. This problem will prepare you for the Multi-Step Test Prep on page 906. Blanca sets up a lens with a focal length f of 15 cm and places a candle 24 cm from the lens. She knows that 1 f = y where is the distance of the object from the lens and y is the distance of the image from the lens. a. Write the equation using the given values. b. For the values of f and given above, how far will the image appear from the lens? c. How will distance between the image and the lens be affected if Blanca uses a lens with a focal length of 18 cm? 904 Chapter 12 Rational Functions and Equations

60 46. Critical Thinking Can you cross multiply to solve all rational equations? If so, eplain. If not, how do you identify which ones can be solved using cross products? 47. Write About It Solve = 3. Eplain each step and why you chose the - 1 method you used. 48. Which value is an etraneous solution to _ _ = 2 _ ? Which is a solution to_ _ 1 = 3_ 2-3? What are the solutions of 5 _ 2 = 1 _ _ 3? -3 and 5-3 and 2 2 and 3 3 and -5 CHALLENGE AND EXTEND 51. Below is a solution to a rational equation. Use an algebraic property to justify each step. Solve 3 _ = 6_ + 4. Statements a. 3( + 4) = 6 Reasons b = 6 c. 12 = 3 d. 4 = 52. For what value of a will the equation a = 7 - a have no solution? 53. Luke, Eddie, and Ryan can do a job in 1 hour and 20 minutes if they work together. Working alone, it takes Ryan 1 hour more to do the job than it takes Luke, and Luke does the job twice as fast as Eddie. How much time would it take each to do the job working alone? SPIRAL REVIEW Identify which lines are parallel and which lines are perpendicular. (Lesson 5-8) 54. y = 1 _ 3 ; y = 3 + 1; y = y =-2; y = 2-2; y = 1 _ y =- - 3; y = - 2; y = y =- 2 _ 3 + 2; y = 3 _ 2 + 3; y =- 3 _ 2-1 Solve each equation. Check your answer. (Lesson 11-9) = = 4 Graph each rational function. (Lesson 12-2) 61. y = _ y = 2_ = 63. y =- 1 _ Solving Rational Equations 905

61 SECTION 12B Operations with Rational Epressions and Equations An Upside-Down World Jamal is studying lenses and their images for a science project. He finds in a science book that a magnifying glass can be used to project upside-down images on a screen. The equation 1 f = y relates the focal length of the lens f, the distance of the object from O the lens, and the distance Object f Image of the image from the lens y. The focal length of Jamal s lens is 10 cm. y 1. Solve the given equation for y using the given value of f. 2. Jamal eperiments with a candle, the lens, and a screen. Given that the focal length remains constant, use a table for the -values 0, 2, 4, 6, 8, 10, 12, 14, and 16 cm. For which -values are the y-values positive? 3. Graph the function y(). Label the aes. Magnification for images is the ratio of the height of the image to the height of the object. This is also equal to the ratio of the distance between the image and the lens and the distance between the object and the lens: M = I O = y. I is the height of the image, O is the height of the object, y is the distance of the image from the lens, and is the distance of the object from the lens. I 4. If the height of a candle is 15 cm and the projected image of that candle is 37.5 cm, what is the magnification of the lens? 5. As Jamal moves the candle further from the lens (increases ), and the distance between the lens and the screen remains the same (y remains constant), does the magnification M increase or decrease? Eplain. 906 Chapter 12 Rational Functions and Equations

62 Quiz for Lessons 12-4 Through 12-7 SECTION 12B 12-4 Multiplying and Dividing Rational Epressions Multiply. Simplify your answer. 1. _ n + 3 (n 2-5n) 2. n _ 5a 2 b 3 5 ab 2a 4 5 _ bc 20c 5. _ 3h 3-6h _ 4g 10g 2 2 gh - 2g 8_ ( ) 4. 6 y 2 _ 2 2 y _ 6 4 y m 2 + m - 2 m 2-2m - 8 m2-8m m - 3 Divide. Simplify your answer. 7. 2_ n _ n n _ b 3 c _ b 2 c (4b 2 + 4b) 12-5 Adding and Subtracting Rational Epressions Add or subtract. Simplify your answer _ 2p - 13 _ 2p _ 2t 4t + _ t _ 3m 2 4m + _ 5m m 5 m 2 - m - 2 m 2 + 6m + 5-2_ 15. m _ - 2 _ _ - 2 _ Julianne competes in a biathlon that consists of a 25-mile running leg and a 45-mile biking leg. Julianne averages 3 times the rate of speed on her bike that she does on her feet. Let r represent Julianne s running rate of speed. Write and simplify an epression, in terms of r, that represents the time it takes for Julianne to complete both legs of the race. Then determine how long it will take Julianne to complete the race if she runs an average of 10 mi/h Dividing Polynomials Divide. 17. (6d 2 + 4d) 2d 18. ( ) (-3 2 ) 19. ( ) (2 + 1) Divide using long division. 20. (a 2 + 3a - 10) (a - 2) 21. (4y 2-9) (2y - 3) 22. ( ) ( + 2) 12-7 Solving Rational Equations Solve. Identify any etraneous solutions. 23. _ 3 = 4_ 24. _ 1-1 = _ _ + _ 4 2 t 3t = 4_ t _ n = _ 7 d n _ d + 8 = _ _ - 6 d = _ It takes Dustin 2 hours to shovel the snow from his driveway and sidewalk. It takes his sister 3 hours to shovel the same area. How long will it take them to shovel the walk if they work together? Ready to Go On? 907

63 EXTENSION Trigonometric Ratios Objectives Find the three basic trigonometric ratios in a right triangle. Use trigonometric ratios to find missing lengths. Vocabulary trigonometric ratios sine cosine tangent A trigonometric ratio is a ratio of the lengths of two sides in a right triangle. Three basic trigonometric ratios are sine, cosine, and tangent, abbreviated sin, cos, and tan, respectively. Trigonometric Ratios sin A = cos A = tan A = leg opposite A = _ a hypotenuse c leg adjacent to A = _ b hypotenuse c leg opposite A side adjacent to A = _ a b EXAMPLE 1 Finding the Value of a Trigonometric Ratio Find each trigonometric ratio to the nearest thousandth. A sin A opposite sin A = hypotenuse = _ 3 = B cos A adjacent cos A = hypotenuse = _ 4 = C tan A tan A = _ opposite adjacent = _ Use the figure above to find sin B, cos B and tan B. EXAMPLE 2 Aviation Application To find trigonometric ratios on a graphing calculator, press,, or and then the value of the degree. Be sure your calculator is in degrees. A plane takes off with a 9 angle of ascent. What is the plane s altitude when it has covered a horizontal distance of 184,800 feet? Round your answer to the nearest foot. Draw a diagram to model the problem. tan A = _ opposite adjacent tan 9 = h_ 184, ,800(tan 9 ) = h Multiply both sides by 184, ,269.4 h Simplify the left side with a calculator. The plane s altitude is about 29,269 feet. 908 Chapter 12 Rational Functions and Equations

64 2. Construction A 14-foot ladder is leaning against a building. The ladder makes a 70 angle with the ground. How far is the base of the ladder from the building? Round your answer to the nearest tenth of a foot. EXTENSION Eercises Use the diagram for Eercises 1 and Find sin A, cos A, and tan A to the nearest thousandth. 2. Find sin B, cos B, and tan B to the nearest thousandth. Find the value of. Round answers to the nearest tenth Diving If a submarine travels 3 miles while rising to the surface at a 9 angle, how deep was the submarine when it started? Round your answer to the nearest tenth of a mile. 7. Construction A wheelchair ramp is to have an angle of 4.5 with the ground. The deck at the top of the ramp is 20 inches above ground level. a. Draw a diagram to illustrate the situation. b. How long should the ramp be? Round your answer to the nearest tenth of an inch. c. How far from the deck should the ramp begin? Round your answer to the nearest tenth of an inch. 8. Navigation The top of a lighthouse is 40 meters above sea level. The angle of elevation from a fishing boat to the top of the lighthouse is 20. How far is the fishing boat from the base of the lighthouse? Round your answer to the nearest tenth of a meter. 9. Write About It What would have to be true about the legs of a right triangle for an angle to have a tangent of 1? What would be the measure of that angle? Etension 909

65 Vocabulary asymptote discontinuous function ecluded value inverse variation rational equation rational epression rational function Complete the sentences below with vocabulary words from the list above. 1. A(n)? is an algebraic epression whose numerator and denominator are polynomials. 2. A function whose rule is a quotient of polynomials in which the denominator has a degree of at least 1 is a(n)?. 3. A(n)? is an equation that contains one or more rational epressions. 4. A(n)? is a relationship that can be written in the form y = _ k, where k is a nonzero constant. 5. A function is a? if its graph contains one or more jumps, breaks, or holes Inverse Variation (pp ) EXAMPLE Write and graph the inverse variation in which y = 2 when = 3. y = k _ Use the form y = k. 2 = _ k 3 Substitute known values. 6 = k Multiply by 3 to find the value of k. y = 6 _ Substitute 6 for k in y = k y und Make a table of values and plot the points. EXERCISES Tell whether each relationship represents an inverse variation. Eplain. 6. y 7. y Write and graph the inverse variation in which y = 4 when = Write and graph the inverse variation in which y = _ 1 when = Let 1 = 5, y 1 =-6, and 2 = 2. Let y vary inversely as. Find y The number of fleet vehicles a town can afford to buy varies inversely as the price of each car. If the town can afford 3 cars priced at $22,000 each, what must the price of a car be in order for the town to purchase 5 of them? 910 Chapter 12 Rational Functions and Equations

66 12-2 Rational Functions (pp ) EXAMPLE Graph the function y = Since the numerator is 1, use the asymptotes and translate y = _ 1. Find the asymptotes. =-1 b =-1 y = 3 c = 3 Graph the asymptotes. Draw smooth curves to show the translation. EXERCISES Identify the ecluded values and the vertical and horizontal asymptotes for each rational function. 12. y = 13. y = y = _ y = 2_ Graph each function. 16. y = _ y = 4_ y = y = A rectangle has an area of 24 c m 2. If represents the width, then y = 24 represents the length y. Describe the reasonable domain and range values and graph the function Simplifying Rational Epressions (pp ) EXAMPLE Simplify the rational epression, if possible. Identify any ecluded values Factor the denominator. ( + 3)( - 1) ( + 3)( - 1) 1 Divide out common factors. ( + 3) Simplify. Identify the ecluded values = 0 To find ecluded values, set the denominator equal to 0. ( + 3)( - 1) = 0 Factor. + 3 = 0 or - 1 = 0 Use the Zero Product Property. = -3 or = 1 Solve each equation for. The ecluded values are -3 and 1. EXERCISES Find any ecluded values of each rational epression _ 5p 23. t_ t 2 - t 25. _ _ r Simplify each rational epression, if possible. Identify any ecluded values r 2 _ 21r k 2 _ 6k 3-9 k 2 _ What is the ratio of the area of the square to the area of the circle? Study Guide: Review 911

67 12-4 Multiplying and Dividing Rational Epressions (pp ) EXAMPLES Multiply. Simplify your answer. _ ( 2-2) _ ( 2-2) _ ( 2-2) ( 2-2) 1 Divide. Simplify your answer. _ 32 2 y 2 _ 2y 2 7z 28z 3 _ 32 2 y 2 28z 3 _ 7z 2 2y 32 _ 16 2 y _ 1 z z y 21 Factor. Divide out common factors. Simplify. Multiply by the reciprocal z 2 Simplify. Divide out common factors. EXERCISES Multiply. Simplify your answer b_ 3b - 6 (b 2 - b - 2) _ 5ab 2ab 3a 2 2 _ b a 2 b 37. _ b + 2 2b b b 2 + 2b - 24 b 2-16 n 2 - n - 12 n 2 + 2n - 24 n 2 + 3n + 2 n 2-4n - 21 Divide. Simplify your answer. 4 _ ( 2-9) _ 3c 2d _ -4c 2 d 8d 2 3b + b 40. _ 2 b + 3 b _ b _ 2 3b + 1 y _ 21 y n 3 _ 4m 2 n _ 2-4 3mn A bag contains red and blue marbles. It has 8 more red marbles than blue marbles. Veronica reaches into the bag and selects one marble at random. She sets the marble aside and then selects another. What is the probability that she selects two red marbles? 12-5 Adding and Subtracting Rational Epressions (pp ) EXAMPLES Add or subtract. Simplify your answer. 7 _ 3y - _ 2-3 3y 7 -( 2-3) 3y y _ y Subtract numerators. Distribute. Simplify. 3w_ w w 2-2w w_ w (w - 5)(w + 3) 3w (w + 3) (w - 5)(w + 3) + 4 (w - 5)(w + 3) 3w 2 + 9w (w - 5)(w + 3) + 4 (w - 5)(w + 3) 3w 2 + 9w + 4 (w - 5)(w + 3) Factor to find the LCD. Write each epression using the LCD. Add and simplify. EXERCISES Find the LCM of the given epressions a 2 2 b, 10ab , 5-15 Add or subtract. Simplify your answer. 47. _ b 2 2b + _ 8 2b 8p 49. p 2-4p _ 3b b - _ 5-2b 7 - b 52. 3_ 5m + _ m m 2 p 2-4p _ _ _ n - 5 n _ n + 5 n _ h 2 + 2h h _ 3h h 54. A scout troop hikes 10 miles to the top of a mountain. Because the return trip is downhill, the troop is able to hike 3 times faster on their way down. Let r represent the troop s rate to the mountaintop. Write and simplify an epression for the round-trip hiking time in terms of r. 912 Chapter 12 Rational Functions and Equations

68 12-6 Dividing Polynomials (pp ) EXAMPLES Divide. ( ) Write as a rational 3 epression. _ _ _ Divide each term 3 separately. 6_ _ _ 21 Divide out common factors Simplify. ( ) ( - 2) ( ) (6 2-12) (17-34) _ 33-2 EXERCISES Divide. 55. (4n 3-6 n 2-10n) 2n 56. ( ) (-5 2 ) _ h h m 2-2m - 24 m Divide using long division. 63. ( ) ( + 3) 64. ( ) ( - 5) 65. (p 2 + 2p - 8) (p + 4) 66. ( ) ( + 2) 67. (2n 2-3n + 1) (n - 5) 68. (3b 3-4b + 2) (b - 2) 69. ( ) ( + 2) 6n 2-13n - 5 2n m 2 + m - 4 m Solving Rational Equations (pp ) EXAMPLE Solve. Identify any etraneous solutions. 3_ = _ 2 ( + 3) ( 3_ ) = ( + 3)2 _ ( + 3) ( 3_ + 3) - ( + 3)4 = ( + 3)2 _ ( + 3) ( 1 3_ 1) ( + 3)4 = 1 ( + 3) _ ( + 3) = 2 ( + 3) = = = (4 + 3)( + 2) = 0 or + 2 = 0 =- 3 _ 4 or =-2-3 or 0, so there are no etraneous solutions. EXERCISES Solve. Identify any etraneous solutions = 3 _ r _ 7 = _ _ 6 b = _ b 74. 2_ - 1 = _ _ _ 4 = _ _ 3b + 4 = _ 1 3b _ = _ _ 3y = _ -2 2 y 75. _ 2 + _ 1 2 = _ _ = _ - 4 = 8_ 2-16 _ _ = 3_ m_ m - 5 = 7-3_ 83. _ - 4 m = _-2-2 Study Guide: Review 913

69 1. Write and graph the inverse variation in which y =-4 when = The number of posters the Spanish Club can buy varies inversely as the cost of each poster. The club can buy 15 posters that cost $2.60 each. How many posters can the club buy if they cost $3.25 each? Identify the ecluded values and the vertical and horizontal asymptotes for each rational function. 3. y = 3_ y = y = Simplify each rational epression, if possible. Identify any ecluded values. 6. 2b_ 4b b 2-2b b Multiply. Simplify your answer. 10. _ ( - 3) 11. 2a 2 2 _ b 5b 3 Divide. Simplify your answer. _ 15a 2 b a _ 4 2 y y _ 12y y 2 _ 3b 2-6b 2b b _ 2b b Add or subtract. Simplify your answer. 16. _ b b _ 5b 3m_ 2m + 2 2m 17. _ _ _ _ 3 - _ y 2 2 _ + 4 y _ y 3 - y Divide. 22. (8t 2-2t) 2t Divide using long division. 24. k 2-2k - 35 k (2w 2 + 5w - 12) (w + 4) 26. ( ) ( + 2) 27. The area of rectangle can be modeled by A() = 3-1. The length is - 1. a. Find a polynomial to represent the width of the rectangle. b. Find the width when is 6 cm. Solve. Identify any etraneous solutions _ - 1 = 9_ _ n - 1 = n_ n _ n + 2 = _ n - 4 n Julio can wash and wa the family car in 2 hours. It takes Leo 3 hours to wash and wa the same car. How long will it take them to wash and wa the car if they work together? 914 Chapter 12 Rational Functions and Equations

70 FOCUS ON SAT MATHEMATICS SUBJECT TESTS The topics covered on each SAT Mathematics Subject Test vary only slightly each time the test is given. Find out the general distribution of test items across topics, and then identify the areas you need to concentrate on while studying. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. To prepare for the SAT Math Subject Tests, start reviewing material several months before your test date. Take sample tests to find the areas you need to focus on. You are not epected to have studied all topics on the test. 1. Which set of ordered pairs satisfies an inverse variation? (A) (6, 3) and (8, 4) (B) (2, -3) and (4, 5) (C) (4, -2) and (-5, 10) (D) (2, 6) and (-3, -4) (E) ( 4, 1 _ 4 ) and ( -4, 1 _ 4 ) 2. If 3_ + 3 = _ 7, what is? 2-9 (A) -12 (B) -3 (C) - 9 _ 4 (D) 9 _ 4 (E) 3 3. What is h if ( h) ( + 1) has a remainder of 15? (A) -10 (B) -5 (C) 5 (D) 10 (E) The graph of which function is shown? (A) f () = 2_ (B) f () = 4_ (C) f () = (D) f () = (E) f () = 4_ _ _ Which function has the same graph as f () = ecept at = 5? (A) g() = _ (B) g() = _ (C) g() = ( - 5)( + 2) ( + 5)( - 1) (D) g() = + 2 ( - 5)( + 1) (E) g() = - 2 College Entrance Eam Practice 915

71 Multiple Choice: Choose Combinations of Answers Some multiple-choice test items require selecting a combination of correct answers. The correct response is the most complete option available. To solve this type of test item, determine if each statement is true or false. Then choose the option that includes each correct statement. Which of the following has an ecluded value of -5? I. II. 5_ ( + 5) I only II and III III. IV. _ _ ( + 2) II, III, and IV III and IV Look at each statement separately and determine whether it is true. You can keep track of which statements are true in a table. Statement I The denominator, - 5, equals 0 when = 5. Statement I does not answer the question, so it is false. Statement II The denominator, 2 ( + 5), equals 0 when =-5. Statement II does answer the question, so it is true. Statement Statement III The denominator, (5 + 25)( + 10), equals 0 when =-5 or =-10. Statement III does answer the question, so it is true. I II III IV True/False False Statement IV The denominator, , can be factored as 2 ( + 5)( + 1). This epression equals 0 when =-5 or =-1. Statement IV does answer the question, so it is true. True True True Statements II, III, and IV are all true. Option C is the correct response because it includes all the true statements. Options B and D contain some of the true statements, but option C is the most complete answer. 916 Chapter 12 Rational Functions and Equations

72 Evaluate all of the statements before deciding on an answer choice. Make a table to keep track of whether each statement is true or false. Read each test item and answer the questions that follow. Item A Which dimensions represent a rectangle that has an area equivalent to the epression ? I. l = + 8 w = 2 ( + 1) II. l = w = III. l = + 2 (2 + 2)( + 4) w = 1 I only III only (3-1) ( ) I and II I, II, and III 1. How do you determine the area of a rectangle? 2. Daisy realized that the area of rectangle I was equivalent to the given area and selected option A as her response. Do you agree? Eplain your reasoning. 3. Write a simplified epression for the width of rectangle II. 4. Eplain each step for determining the area of rectangle III. 5. If rectangle II has an area equivalent to the given epression, then which options can you eliminate? Item B Which epression is undefined for = 3 or = -2? I ( - 3)( + 6) II. _ III. IV. ( - 2) ( + 2)( - 1) 14_ I, III, and IV III and IV I and II I and IV 6. When is an epression undefined? 7. Henry determined that statement I is undefined when = 3. He decides it is an incorrect answer because the epression is defined when =-2. Should he select option H by process of elimination? Eplain your reasoning. 8. Make a table to determine the correct response. Item C Which rational function has a graph with a horizontal asymptote of y = 4? I. y = _ -4 III. y = - 4 II. y = 1 _ + 4 IV. y =- 1 _ + 4 I and III II only I and II II and IV 9. Where does the horizontal asymptote of the function in statement I occur? 10. Using your answer from Problem 9, which option(s) can you eliminate? Eplain your reasoning. 11. Look at the options remaining. Which statement would be best to check net? Eplain your reasoning. Test Tackler 917

73 KEYWORD: MA7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1 12 Multiple Choice 1. Simplify the epression 4 (2d - 1) - 6d. 2d -4d + 3 2d - 4 2d Which equation is the result of solving 3 + 2y = 8 for y? y = 3 _ 2-4 y = y = Which function is shown in the graph? y = 3 y = _ 3 y = + 3 y = 1 _ + 3 y =- 3 _ The drama club needs to raise at least $1400 for a field trip. The club was given $150 by the school administration. Club members are selling key chains for $5 each. Which inequality represents the number of key chains k that the drama club needs to sell to go on its field trip? k k k k Which epression is NOT equivalent to _ _ _ - 1 3_ - 1? 6. Which situation best describes a negative correlation? The intensity level of an eercise and the number of Calories burned per minute The amount of time that an electronic game is on and the amount of power remaining in the game s batteries The height of a tree and the amount of ink in a ballpoint pen The daytime temperature and the number of people at an ice cream stand 7. Which epression is equivalent to _ 3m 2 n 5m 12m 2 _ n 4 12m 2 n 3 12m 3 _ n 3 12m _ n 8. Simplify 3 _ + 3 _ 5. 18_ 5 9_ Which of the following has a slope of -3? 10. What is ( ) (-4 2 )? _ _ _ 20mn? n Chapter 12 Rational Functions and Equations

74 11. Abe flips two coins. What is the probability of both coins landing heads up? Which is a solution to 8 16 n_ n + 2 = _ -8 n? Which of the following is equivalent to _ ( 2 5 y 2 8 ) -2? 16_ 4_ 8 y 4 4 y 2 _ 8 5 _ 16y What situation can be modeled by the function y = _ 140? The cost of attending a ski trip is $140 for each person who attends. The area of a rectangle with a width of 140 meters varies directly as its length. The cost per person of a boat rental is $140 divided by the number of people. Attendance at this year s concert was 140 people more than at last year s concert. Know the rules for writing gridded-response answers. For eample, can you write the fraction 2 as or as 0.67? Different tests 3 may have different rules, so pay close attention to the directions. Gridded Response 15. What is the ecluded value for the rational epression _ ? Short Response 20. Mr. Lui wrote 15-5 on the board a. Eplain what kind of epression it is. b. Simplify the epression. Show your work. c. Identify any ecluded values. 21. Describe the similarities and differences between the graph of f () = and the graph of g() = 1 _ What are 2 values of b that will make b - 20 factorable? Eplain your answer. 23. Brandee makes an hourly wage. In the last pay period, she earned $800 for regular hours and $240 for overtime hours. Her overtime rate of pay is 50% more per hour than her regular rate of pay r. Write and simplify an epression, in terms of r, that represents the number of hours h Brandee worked in the pay period. Show your work. Etended Response 24. Principal Farley has $200 to pay for some teachers to attend a technology conference. The company hosting the conference is allowing 2 teachers to attend for free. The number of teachers y that can be sent to the conference is given by the function y = _ , where is the cost per teacher. a. Describe the reasonable domain and range values for this function. b. Identify the vertical and horizontal asymptotes. c. Graph the function. d. Give two whole-number solutions to the equation and describe what they mean in the contet of this situation. 16. What is the net term in the geometric sequence 2000, 1600, 1280, 1024,? 17. What is the constant of variation if y varies inversely as and y = 3 when = 6? 18. What is the value of 4 0 -(2-3 )? 19. Identify the ecluded value for y = - 4 _ - 2. Cumulative Assessment, Chapters

75 OHIO New Bremen Cincinnati Cincinnati Reds Organized in 1866, the Cincinnati Reds, known then as the Cincinnati Red Stockings, were the first major league baseball team. They now play in a stadium built in 2003 called the Great American Ballpark. Choose one or more strategies to solve each problem. 1. The table shows the total payroll for the Reds from 2000 to What percent did the total payroll increase from 2000 to 2005? 2. Assume the payroll percent increase for the net years is the same as from 2004 to Write an eponential growth function to model this situation. 3. Using the function found in Problem 2, what is the epected total payroll for the Reds in the year 2010? 4. Suppose that at a Cincinnati Reds game, tickets in the blue zone average $28 and tickets in the green zone average $20. Suppose twice as many tickets were sold to fans in the green zone as in the blue zone and the total ticket sales for these two zones was $204,000. Fans bought 14,197 tickets in other zones and sections. How many people had tickets to the game? Payroll for The Cincinnati Reds Year Total Payroll ($) ,217, ,784, ,050, ,355, ,615, ,892, Chapter 12 Rational Functions and Equations

76 Bicycle Museum of America The Bicycle Museum of America in New Bremen, Ohio, is home to one of the largest private collections of bicycles and bicycle memorabilia in the world. The collection represents every era, including antique bicycles from the 1800s, balloon-tire classics of the 1940s and 1950s, and banana-seat bikes with high-rise handlebars from the 1960s. Choose one or more strategies to solve each problem. 1. Alfred Letourner was one of the top racers of his era. On May 17, 1941, at an event sponsored by Schwinn, Letourner rode a bike similar to the one in the photo with a gear ratio of 9 1 to 1. The bike only weighed 20 2 pounds. At this event, Letourner shattered speed records when he rode a mile in seconds. At this pace, about how many miles per hour was Letourner riding? The table shows the relationship between the number of teeth on the pedal gear and the number of teeth on the wheel gear of a bicycle. This relationship affects the speed and effort of pedaling. For eample, if the pedal gear has 39 teeth and the wheel gear has 17 teeth, then the gear ratio is 39 = This number represents 17 the number of turns of the wheel for every full turn of the pedal. 2. Which gear combination shown in the table would yield the highest number of rotations of the tires to the turn of the pedal? About how many turns of the wheel for every full turn of the pedal can be epected with this combination? 3. If a mountain bike has tires with a diameter of 26 inches, how far, in feet, will a rider travel for each full turn of the pedal if the pedal gear has 39 teeth and the wheel gear has 14 teeth? Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Problem Solving on Location 921